Method, apparatus and medium for decoding or encoding

ABSTRACT

A method of utilizing an 8-bit primary transform core matrix to decode compressed video or image data or encode uncompressed video or image data. The method may include determining whether to use a first transform core matrix that is of a first size type or a second transform core matrix that is of a second size type that is smaller than the first size type. When a result of the determination is to use the first transform core matrix, encoding or decoding the target file using the 8-bit primary transform core matrix that has the size that is 64-point or larger. If not, the method may include extracting the second transform core matrix of the second size type from the first transform matrix and encoding or decoding the target file using the extracted second transform core matrix.

PRIORITY INFORMATION

This is a continuation of U.S. application Ser. No. 16/225,982, filedDec. 19, 2018, which claims priority from U.S. Provisional ApplicationNo. 62/692,658, filed on Jun. 29, 2018, in the United States Patent &Trademark Office, the disclosures of which are incorporated herein byreference in their entireties.

BACKGROUND Field

This disclosure is directed towards video compression technologies. Inparticular, the present disclosure is directed towards next-generationvideo coding technologies including video coding/decoding technologiesbeyond High Efficiency Video Coding (HEVC), such as Versatile VideoCoding (VVC). More specifically, an aspect of the disclosure is directedtowards a VVC primary transform method, device and computer medium thatuse a large, 8-bit transform core having a size that is equal to orgreater than 64×64.

Description of Related Art

Video coding and decoding using inter-picture prediction with motioncompensation has been known for decades. Uncompressed digital video canconsist of a series of pictures, each picture having a spatial dimensionof, for example, 1920×1080 luminance samples and associated chrominancesamples. The series of pictures can have a fixed or variable picturerate (informally also known as frame rate), of, for example 60 picturesper second or 60 hertz (Hz). Uncompressed video has significant bitraterequirements. For example, 1080p60 4:2:0 video at 8 bit per sample(1920×1080 luminance sample resolution at 60 Hz frame rate) requiresclose to 1.5 Gbit/s bandwidth. An hour of such video requires more than600 GByte of storage space.

One purpose of video coding and decoding can be the reduction ofredundancy in the input video signal, through compression. Compressioncan help reduce aforementioned bandwidth or storage space requirements,in some cases by two orders of magnitude or more. Both lossless andlossy compression, as well as a combination thereof can be employed.Lossless compression refers to techniques where an exact copy of theoriginal signal can be reconstructed from the compressed originalsignal. When using lossy compression, the reconstructed signal may notbe identical to the original signal, but the distortion between originaland reconstructed signal is small enough to make the reconstructedsignal useful for the intended application. In the case of video, lossycompression is widely employed. The amount of distortion tolerateddepends on the application; for example, users of certain consumerstreaming applications may tolerate higher distortion than users oftelevision contribution applications. The compression ratio achievablecan reflect that: higher allowable/tolerable distortion can yield highercompression ratios.

A person of ordinary skill generally understands videocompression/decompression technology. In general, to compress video orimage data, a series of functional steps are performed to result in acompressed video or image file. Although an image, such as a 360° image(e.g., captured by a 360° camera) may be suitable for compression, forease of reading, compression of a video file will be explained. Togenerate a compressed video file, under conventional standards (e.g.,H.264, H.265) an uncompressed video sample stream that is received froma video source may be partitioned or parsed, which results in a block ofsamples of two or more reference pictures.

Bi-Prediction can relate to techniques where a prediction unit (PU),such as a predicted block of samples, can be predicted from two motioncompensated blocks of samples of two or more reference pictures.Bi-prediction was first introduced into video coding standards inMPEG-1, and has also been included in other video coding technologiesand standards such as MPEG-2 Part 2 (or H.262), H.264 and H.265 as well.

When decompressing a compressed video file, during the reconstruction ofa sample of a bi-predicted PU, motion compensated and interpolated inputsamples from each reference block can be multiplied by a weightingfactor that can be different for each reference block, and such weightedsample values of the two reference blocks can be added to generate thesample under reconstruction. Such sample can be processed further bymechanisms such as loop filtering.

In MPEG-1 and MPEG-2, the weighting factors can be determined based onthe relative temporal distance between the picture to which the PU underreconstruction belongs to, and the two reference pictures. This ispossible because, in MPEG-1 and MPEG-2, one of the two reference I or Ppictures was in the “past”, and the other in the “future” (in terms ofpresentation order) of the B-picture under reconstruction, and becausein MPEG-1 and MPEG-2, there was a well-defined timing relationshipestablished for any picture under reconstruction in relation to itsreference pictures.

Starting with H.264, the reference picture selection concepts forbi-predicted pictures were relaxed such that the reference pictures onlyneeded to be earlier in decoding order, but not in presentation order.Further, the notion of time was also relaxed in that neither H.264 norH.265 require a constrained/fixed picture interval in the time domain.Therefore, a decoder cannot calculate weighting factors any more basedon the timing information available in the bitstream. Instead, H.264 andH.265 include a “default” of 0.5 as the weighting factor for thereference samples of a bi-predicted picture. This default can beoverwritten by a syntax available in the slice header known aspred_weight_table( ). The default of 0.5 or the information in thepred_weight_table may apply to all bi-predicted PUs in a given slice.

Non-Patent Literature 1 shows the H.265/HEVC standard. However, a needfor standardization of future video coding technology with a compressioncapability that significantly exceeds that of the HEVC standard(including its extensions) has been studied by the inventors.

Non-Patent Literature 2 discloses a recently launched standardizationformat for next-generation video coding beyond HEVC called VersatileVideo Coding (VVC) with a version VTM (VVC Test Model). VVC maygenerally provide a large (e.g., 64-point or higher) transform coreusing a 10-bit integer matrix.

LISTING OF RELATED ART

-   Non-Patent Literature 1: “Transform design for HEVC with 16 bit    intermediate data representation”; published by: Joint Collaborative    Team on Video Coding (JCT-VC) of ITU-T SG16 WP3 and ISO/IEC    JTC1/SC29/WG11, 5th Meeting: Geneva, CH, Mar. 16-23, 2011 (Document:    JCTVC-E243). The H.265/HEVC (High Efficiency Video Coding) standard    may include Version 1 (2013), Version 2 (2014), Version 3 (2015) and    Version 4 (2016) published by ITU-T VCEG (Q6/16) ISO/IEC MPEG (JTC    1/SC 29/WG 11).-   Non-Patent Literature 2: “Algorithm description for Versatile Video    Coding and Test Model 1 (VTM 1)”; published by Joint Video Experts    Team (WET) of ITU-T SG 16 WP 3 and ISO/IEC JTC 1/SC 29/WG 11, 10th    Meeting: San Diego, US, Apr. 10-20, 2018 (Document: WET-J1002-v2).

Problem(s) to be Solved

However, a need exists for an 8-bit, large (e.g., 64-point or higher)core in the VVC to reduce complexity, reduce memory for storingtransform cores and/or reduce input bit-width of multiplier/add/suboperations. In general, the 2-, 4-, 8-, 16- and 32-point DCT-2 transformcore, which are 8-bit, cannot be extracted from, for example, a 64-pointDCT-2 transform core, which is 10-bit in conventional VVC, thusinevitably increasing the cost of applying primary transform because aseparate 64-point transform core would need extra storage and logic forimplementation. Comparing to 10-bit primary transform cores, the 8-bitprimary transform core reduces the complexity because of the reducedmemory for storing transform cores and reduced input bit-width ofmultiplier/add/sub operations.

SUMMARY

According to an aspect of the disclosure, a method of utilizing an 8-bitprimary transform core matrix to decode compressed video or image dataor encode uncompressed video or image data may include: receivinginformation regarding a target file for encoding or decoding, the targetfile for encoding or decoding being one of: a compressed video or imagedata file or an uncompressed video or image data file, and determining,for the encoding or decoding of the target file, whether to use a firsttransform core matrix that is of a first size type or a second transformcore matrix that is of a second size type that is smaller than the firstsize type, wherein the first size type is the 8-bit primary transformcore matrix, which has the core matrix that is represented by 8-bits andthat has a size that is 64-point or larger. The method according to theaspect of the disclosure may further include: when a result of thedetermination is to use the first transform core matrix of the firstsize type, causing or transmitting information that causes the targetfile for encoding or decoding to be encoded or decoded using the 8-bitprimary transform core matrix that has the size that is 64-point orlarger, and when a result of the determination is to use the secondtransform core matrix of the second size type that is smaller than thefirst size type: extracting the second transform core matrix of thesecond size type from the 8-bit primary transform core matrix that is64-point or larger and causing or transmitting information that causesthe target file for encoding or decoding to be encoded or decoded usingthe extracted second transform core matrix of the second type.

According to an aspect of the disclosure, an apparatus for utilizing an8-bit primary transform core matrix to decode compressed video or imagedata or encode uncompressed video or image data may include: at leastone memory configured to store computer program code, and at least oneprocessor configured to access the at least one memory and operateaccording to the computer program code. The computer program code maycomprise first applying code, second applying code, third applying codeand fourth applying code.

The first applying code may be configured to cause the at least oneprocessor to receive information regarding a target file for encoding ordecoding. The target file for encoding or decoding may be one of: acompressed video or image data file or an uncompressed video or imagedata file. The second applying code may be configured to cause the atleast one processor to determine, for the encoding or decoding of thetarget file, whether to use a first transform core matrix that is of afirst size type or a second transform core matrix that is of a secondsize type that is smaller than the first size type, where the first sizetype is the 8-bit primary transform core matrix, which has the corematrix that is represented by 8-bits and that has a size that is64-point or larger.

The third applying code may be configured cause the at least oneprocessor to when a result of the determination is to use the firsttransform core matrix of the first size type, cause or transmitinformation that causes the target file for encoding or decoding to beencoded or decoded using the 8-bit primary transform core matrix thathas the size that is 64-point or larger. The fourth applying codeconfigured to cause the at least one processor to when a result of thedetermination is to use the second transform core matrix of the secondsize type that is smaller than the first size type: extract the secondtransform core matrix of the second size type from the 8-bit primarytransform core matrix that is 64-point or larger and cause or transmitinformation that causes the target file for encoding or decoding to beencoded or decoded using the extracted second transform core matrix ofthe second type.

According to an aspect of the disclosure, a non-transitorycomputer-readable storage medium storing may store instructions thatcause one or more processors to: receive information regarding a targetfile for encoding or decoding, the target file for encoding or decodingbeing one of: a compressed video or image data file or an uncompressedvideo or image data file; and determine, for the encoding or decoding ofthe target file, whether to use a first transform core matrix that is ofa first size type or a second transform core matrix that is of a secondsize type that is smaller than the first size type, wherein the firstsize type is the 8-bit primary transform core matrix, which has the corematrix that is represented by 8-bits and that has a size that is64-point or larger.

The stored instructions of the non-transitory computer-readable storagemedium may further cause the one or more processors to: when a result ofthe determination is to use the first transform core matrix of the firstsize type, cause or transmit information that causes the target file forencoding or decoding to be encoded or decoded using the 8-bit primarytransform core matrix that has the size that is 64-point or larger; andwhen a result of the determination is to use the second transform corematrix of the second size type that is smaller than the first size type:extract the second transform core matrix of the second size type fromthe 8-bit primary transform core matrix that is 64-point or larger andcause or transmit information that causes the target file for encodingor decoding to be encoded or decoded using the extracted secondtransform core matrix of the second type.

BRIEF DESCRIPTION OF THE DRAWINGS

Further features, the nature, and various advantages of the disclosedsubject matter will be more apparent from the following detaileddescription and the accompanying drawings in which:

FIG. 1 is a schematic illustration of a simplified block diagram of acommunication system in accordance with an embodiment.

FIG. 2 is a schematic illustration of a simplified block diagram of astreaming system in accordance with an embodiment.

FIG. 3 is a schematic illustration of a simplified block diagram of adecoder in accordance with an embodiment.

FIG. 4 is a schematic illustration of a simplified block diagram of anencoder in accordance with an embodiment.

FIG. 5 is a schematic illustration of an exemplary method of encoding ordecoding in accordance with an embodiment.

FIG. 6 is a schematic illustration of outputting a transform core matrixin accordance with an embodiment.

FIG. 7 is a schematic illustration of a computer system in accordancewith an embodiment.

DETAILED DESCRIPTION

FIG. 1 illustrates a simplified block diagram of a communication system(100) according to an embodiment of the present disclosure. The system(100) may include at least two terminals (110, 120) interconnected via anetwork (150). For unidirectional transmission of data, a first terminal(110) may code video data at a local location for transmission to theother terminal (120) via the network (150). The second terminal (120)may receive the coded video data of the other terminal from the network(150), decode the coded data and display the recovered video data.Unidirectional data transmission may be common in media servingapplications and the like.

FIG. 1 illustrates a second pair of terminals (130, 140) provided tosupport bidirectional transmission of coded video that may occur, forexample, during videoconferencing. For bidirectional transmission ofdata, each terminal (130, 140) may code video data captured at a locallocation for transmission to the other terminal via the network (150).Each terminal (130, 140) also may receive the coded video datatransmitted by the other terminal, may decode the coded data and maydisplay the recovered video data at a local display device.

In FIG. 1, the terminals (110-140) may be illustrated as a laptop 110, aserver 120, and smartphones 130 and 140, but the principles of thepresent disclosure are not so limited. Embodiments of the presentdisclosure find application with other equipment including but notlimited to laptop computers, tablet computers, media players and/ordedicated video conferencing equipment. The network (150) represents anynumber of networks that convey coded video data among the terminals(110-140), including for example wireline and/or wireless communicationnetworks. The communication network (150) may exchange data incircuit-switched and/or packet-switched channels. Representativenetworks include telecommunications networks, local area networks, widearea networks and/or the Internet. For the purposes of the presentdiscussion, the architecture and topology of the network (150) may beimmaterial to the operation of the present disclosure unless explainedherein below.

FIG. 2 illustrates, as an example for an application for the disclosedsubject matter, the placement of a video encoder and decoder in astreaming environment. The disclosed subject matter can be equallyapplicable to other video enabled applications, including, for example,video conferencing, digital TV, storing of compressed video on digitalmedia including CD, DVD, memory stick and the like, and so on.

A streaming system may include a capture subsystem (213), which caninclude a video source (201), for example a digital camera, which maycreate an uncompressed video sample stream (202). That sample stream(202), depicted as a bold line to emphasize a high data volume whencompared to encoded video bit streams, can be processed by an encoder(203) coupled to the camera (201). The encoder (203) can includehardware (e.g., a processor or circuit and memory), software, or acombination thereof to enable or implement aspects of the disclosedsubject matter as described in more detail below. The encoded video bitstream (204), depicted as a thin line to emphasize the lower data volumewhen compared to the sample stream, can be stored on a streaming server(205) for future use. One or more streaming clients (206, 208) canaccess the streaming server (205) to retrieve copies (207, 209) of theencoded video bitstream (204). A client (206) can include a videodecoder (210) which decodes the incoming copy of the encoded videobitstream (207) and creates an outgoing video sample stream (211) thatcan be rendered on a display (212) or other rendering device (notdepicted).

FIG. 3 illustrates a functional block diagram of a video decoder (210)according to an embodiment of the present disclosure. As shown in FIG.3, a receiver (310) may receive one or more codec video sequences to bedecoded by the decoder (210). In the same or another embodiment, onecoded video sequence may be received at a time, where the decoding ofeach coded video sequence is independent from other coded videosequences. The coded video sequence may be received from a channel(312), which may be a hardware/software link to a storage device whichstores the encoded video data. The receiver (310) may receive theencoded video data with other data, for example, coded audio data and/orancillary data streams, that may be forwarded to their respective usingentities (not depicted). The receiver (310) may separate the coded videosequence from the other data. To combat network jitter, a buffer memory(315) may be coupled in between receiver (310) and entropydecoder/parser (320) (“parser” henceforth). When receiver (310) isreceiving data from a store/forward device of sufficient bandwidth andcontrollability, or from an isosychronous network, the buffer (315) maynot be needed, or can be small. For use on best effort packet networkssuch as the Internet, the buffer (315) may be required, which can becomparatively large and can advantageously be of adaptive size.

The video decoder (310) may include a parser (320) to reconstructsymbols (321) from the entropy coded video sequence. Categories of thosesymbols include information used to manage operation of the decoder(210), and potentially information to control a rendering device such asa display (212) that is not an integral part of the decoder but can becoupled to it, as was shown in FIGS. 2 and 3. The control informationfor the rendering device(s) may be in the form of SupplementaryEnhancement Information (SEI messages) or Video Usability Information(VUI) parameter set fragments (not depicted). The parser (320) mayparse/entropy-decode the coded video sequence received. The coding ofthe coded video sequence can be in accordance with a video codingtechnology or standard, and can follow principles well known to a personskilled in the art, including variable length coding, Huffman coding,arithmetic coding with or without context sensitivity, and so forth. Theparser (320) may extract from the coded video sequence, a set ofsubgroup parameters for at least one of the subgroups of pixels in thevideo decoder, based upon at least one parameters corresponding to thegroup. Subgroups can include Groups of Pictures (GOPs), pictures, tiles,slices, macroblocks, Coding Units (CUs), blocks, Transform Units (TUs),Prediction Units (PUs) and so forth. The entropy decoder/parser may alsoextract from the coded video sequence information such as transformcoefficients, quantizer parameter values, motion vectors, and so forth.

The parser (320) may perform entropy decoding/parsing operation on thevideo sequence received from the buffer (315), so as to create symbols(321). Reconstruction of the symbols (321) can involve multipledifferent units depending on the type of the coded video picture orparts thereof (such as: inter and intra picture, inter and intra block),and other factors. Which units are involved, and how, can be controlledby the subgroup control information that was parsed from the coded videosequence by the parser (320). The flow of such subgroup controlinformation between the parser (320) and the multiple units below is notdepicted for clarity.

Beyond the functional blocks already mentioned, decoder 210 can beconceptually subdivided into a number of functional units as describedbelow. In a practical implementation operating under commercialconstraints, many of these units interact closely with each other andcan, at least partly, be integrated into each other. However, for thepurpose of describing the disclosed subject matter, the conceptualsubdivision into the functional units below is appropriate.

A first unit may be the scaler/inverse transform unit (351). Thescaler/inverse transform unit (351) may receive a quantized transformcoefficient as well as control information, including which transform touse, a block size, a quantization factor, quantization scaling matrices,etc. as symbol(s) (321) from the parser (320). It can output blockscomprising sample values that can be input into an aggregator (355).

In some cases, the output samples of the scaler/inverse transform (351)can pertain to an intra-coded block, which is a block that is not usingpredictive information from previously reconstructed pictures, but canuse predictive information from previously reconstructed parts of thecurrent picture. Such predictive information can be provided by an intrapicture prediction unit (352). In some cases, the intra pictureprediction unit (352) generates a block of the same size and shape ofthe block under reconstruction, using surrounding already reconstructedinformation fetched from the current (partly reconstructed) picture(356). The aggregator (355), in some cases, adds, on a per sample basis,the prediction information the intra prediction unit (352) has generatedto the output sample information as provided by the scaler/inversetransform unit (351).

In other cases, the output samples of the scaler/inverse transform unit(351) can pertain to an inter-coded block, and a potentially motioncompensated block. In such a case, a Motion Compensation Prediction unit(353) can access reference a picture memory (357) to fetch samples usedfor prediction. After motion compensating the fetched samples inaccordance with the symbols (321) pertaining to the block, these samplescan be added by the aggregator (355) to the output of the scaler/inversetransform unit (in this case called the residual samples or residualsignal) so to generate output sample information. The addresses withinthe reference picture memory form where the motion compensation unitfetches prediction samples can be controlled by motion vectors,available to the motion compensation unit in the form of symbols (321)that can have, for example X, Y, and reference picture components.Motion compensation also can include interpolation of sample values asfetched from the reference picture memory when sub-sample exact motionvectors are in use, motion vector prediction mechanisms, and so forth.

The output samples of the aggregator (355) can be subject to variousloop filtering techniques in the loop filter unit (356). Videocompression technologies can include in-loop filter technologies thatare controlled by parameters included in the coded video bitstream andmade available to the loop filter unit (356) as symbols (321) from theparser (320), but can also be responsive to meta-information obtainedduring the decoding of previous (in decoding order) parts of the codedpicture or coded video sequence, as well as responsive to previouslyreconstructed and loop-filtered sample values.

The output of the loop filter unit (356) can be a sample stream that canbe output to the render device (212) as well as stored in the referencepicture memory (356) for use in future inter-picture prediction.

Certain coded pictures, once fully reconstructed, can be used asreference pictures for future prediction. Once a coded picture is fullyreconstructed and the coded picture has been identified as a referencepicture (by, for example, parser (320)), the current reference picture(356) can become part of the reference picture buffer (357), and a freshcurrent picture memory can be reallocated before commencing thereconstruction of the following coded picture.

The video decoder 320 may perform decoding operations according to apredetermined video compression technology that may be documented in astandard, such as ITU-T Rec. H.265. The coded video sequence may conformto a syntax specified by the video compression technology or standardbeing used, in the sense that it adheres to the syntax of the videocompression technology or standard, as specified in the videocompression technology document or standard and specifically in theprofiles document therein. Also necessary for compliance can be that thecomplexity of the coded video sequence is within bounds as defined bythe level of the video compression technology or standard. In somecases, levels restrict the maximum picture size, maximum frame rate,maximum reconstruction sample rate (measured in, for example megasamplesper second), and/or maximum reference picture size. Limits set by levelscan, in some cases, be further restricted through Hypothetical ReferenceDecoder (HRD) specifications and metadata for HRD buffer managementsignaled in the coded video sequence.

In an embodiment, the receiver (310) may receive additional (redundant)data with the encoded video. The additional data may be included as partof the coded video sequence(s). The additional data may be used by thevideo decoder (320) to properly decode the data and/or to moreaccurately reconstruct the original video data. Additional data can bein the form of, for example, temporal, spatial, or SNR enhancementlayers, redundant slices, redundant pictures, forward error correctioncodes, and so on.

FIG. 4 may be a functional block diagram of a video encoder (203)according to an embodiment of the present disclosure.

The encoder (203) may receive video samples from a video source (201)(that is not part of the encoder) that may capture video image(s) to becoded by the encoder (203).

The video source (201) may provide the source video sequence to be codedby the encoder (203) in the form of a digital video sample stream thatcan be of any suitable bit depth (for example: 8 bit, 10 bit, 12 bit, .. . ), any colorspace (for example, BT.601 Y CrCB, RGB, . . . ) and anysuitable sampling structure (for example Y CrCb 4:2:0, Y CrCb 4:4:4). Ina media serving system, the video source (201) may be a storage device,which may include a memory and processor, that stores previouslyprepared video. In a videoconferencing system, the video source (203)may include a camera that captures local image information as a videosequence. Video data may be provided as a plurality of individualpictures that impart motion when viewed in sequence. The picturesthemselves may be organized as a spatial array of pixels, wherein eachpixel can comprise one or more sample depending on the samplingstructure, color space, etc. in use. A person skilled in the art canreadily understand the relationship between pixels and samples. Thedescription below focusses on samples.

According to an embodiment, the encoder (203) may code and compress thepictures of the source video sequence into a coded video sequence (443)in real time or under any other time constraints as required by theapplication. Enforcing appropriate coding speed is one function ofcontroller (450). Controller controls other functional units asdescribed below and is functionally coupled to these units. The couplingis not depicted for clarity. Parameters set by the controller caninclude rate control related parameters (e.g., picture skip, quantizer,lambda value of rate-distortion optimization techniques), a picturesize, a group of pictures (GOP) layout, a maximum motion vector searchrange, and so forth. A person skilled in the art can readily identifyother functions of controller (450) as they may pertain to video encoder(203) optimized for a certain system design.

Some video encoders operate in what a person skilled in the art readilyrecognizes as a “coding loop.” As an oversimplified description, acoding loop can consist of the encoding part of an encoder (430)(“source coder” henceforth), which is responsible for creating symbolsbased on an input picture to be coded, and a reference picture(s), and a(local) decoder (433) embedded in the encoder (203) that reconstructsthe symbols to create the sample data a (remote) decoder also wouldcreate (as any compression between symbols and coded video bitstream islossless in the video compression technologies considered in thedisclosed subject matter). That reconstructed sample stream is input tothe reference picture memory (434). As the decoding of a symbol streamleads to bit-exact results independent of a decoder location (local orremote), the reference picture buffer content is also bit exact betweenlocal encoder and remote encoder. In other words, the prediction part ofan encoder “sees,” as reference picture samples, exactly the same samplevalues as a decoder would “see” when using prediction during decoding.This fundamental principle of reference picture synchronicity (andresulting drift, if synchronicity cannot be maintained, for example,because of channel errors) is well known to a person skilled in the art.

The operation of the “local” decoder (433) can be the same as of a“remote” decoder (210), which has already been described in detail abovein conjunction with FIG. 3. Briefly referring also to FIG. 3, however,as symbols are available and en/decoding of symbols to a coded videosequence by entropy coder (445) and parser (320) can be lossless, theentropy decoding parts of decoder (210) (including channel (312),receiver (310), buffer (315), and parser (320)) may not be fullyimplemented in local decoder (433).

An observation that can be made at this point is that any decodertechnology except the parsing/entropy decoding that is present in adecoder also necessarily needs to be present, in substantially identicalfunctional form, in a corresponding encoder. The description of encodertechnologies are the inverse of the described decoder technologies. Onlyin certain areas a more detail description is required and providedbelow.

As part of its operation, the source coder (430) may perform motioncompensated predictive coding, which codes an input frame predictivelywith reference to one or more previously-coded frames from the videosequence that were designated as “reference frames.” In this manner, thecoding engine (432) codes differences between pixel blocks of an inputframe and pixel blocks of reference frame(s) that may be selected asprediction reference(s) to the input frame.

The local video decoder (433) may decode coded video data of frames thatmay be designated as reference frames, based on symbols created by thesource coder (430). Operations of the coding engine (432) mayadvantageously be lossy processes. When the coded video data may bedecoded at a video decoder (not shown in FIG. 4), the reconstructedvideo sequence typically may be a replica of the source video sequencewith some errors. The local video decoder (433) replicates decodingprocesses that may be performed by the video decoder on reference framesand may cause reconstructed reference frames to be stored in thereference picture cache (434). In this manner, the encoder (203) maystore copies of reconstructed reference frames locally that have commoncontent as the reconstructed reference frames that will be obtained by afar-end video decoder (absent transmission errors).

The predictor (435) may perform prediction searches for the codingengine (432). That is, for a new frame to be coded, the predictor (435)may search the reference picture memory (434) for sample data (ascandidate reference pixel blocks) or certain metadata such as referencepicture motion vectors, block shapes, and so on, that may serve as anappropriate prediction reference for the new pictures. The predictor(435) may operate on a sample block-by-pixel block basis to findappropriate prediction references. In some cases, as determined bysearch results obtained by the predictor (435), an input picture mayhave prediction references drawn from multiple reference pictures storedin the reference picture memory (434).

The controller (450) may manage coding operations of the video coder(430), including, for example, setting of parameters and subgroupparameters used for encoding the video data.

Output of all aforementioned functional units may be subjected toentropy coding in the entropy coder (445). The entropy coder maytranslate the symbols as generated by the various functional units intoa coded video sequence, by loss-less compressing the symbols accordingto technologies known to a person skilled in the art as, for exampleHuffman coding, variable length coding, arithmetic coding, and so forth.

The transmitter (440) may buffer the coded video sequence(s) as createdby the entropy coder (445) to prepare it for transmission via acommunication channel (460), which may be a hardware/software link to astorage device which would store the encoded video data. The transmitter(440) may merge coded video data from the video coder (430) with otherdata to be transmitted, for example, coded audio data and/or ancillarydata streams (sources not shown).

The controller (450) may manage operation of the encoder (203). Duringcoding, the controller (450) may assign to each coded picture a certaincoded picture type, which may affect the coding techniques that may beapplied to the respective picture. For example, pictures often may beassigned as one of the following frame types from among, for example, anIntra Picture (I picture), a Predictive picture (P picture) and aBi-directionally Predictive Picture (B Picture).

An Intra Picture (I picture) may be one that may be coded and decodedwithout using any other frame in the sequence as a source of prediction.Some video codecs allow for different types of Intra pictures,including, for example Independent Decoder Refresh Pictures. A personskilled in the art is aware of those variants of I pictures and theirrespective applications and features.

A Predictive Picture (P picture) may be one that may be coded anddecoded using intra prediction or inter prediction using at most onemotion vector and reference index to predict the sample values of eachblock.

A Bi-directionally Predictive Picture (B Picture) may be one that may becoded and decoded using intra prediction or inter prediction using atmost two motion vectors and reference indices to predict the samplevalues of each block. Similarly, multiple-predictive pictures can usemore than two reference pictures and associated metadata for thereconstruction of a single block.

Source pictures commonly may be subdivided spatially into a plurality ofsample blocks (for example, blocks of 4×4, 8×8, 4×8, or 16×16 sampleseach) and coded on a block-by-block basis. Blocks may be codedpredictively with reference to other (already coded) blocks asdetermined by the coding assignment applied to the blocks' respectivepictures. For example, blocks of I pictures may be codednon-predictively or they may be coded predictively with reference toalready coded blocks of the same picture (spatial prediction or intraprediction). Pixel blocks of P pictures may be coded non-predictively,via spatial prediction or via temporal prediction with reference to onepreviously coded reference pictures. Blocks of B pictures may be codednon-predictively, via spatial prediction or via temporal prediction withreference to one or two previously coded reference pictures.

The video coder (203) may perform coding operations according to apredetermined video coding technology or standard, such as ITU-T Rec.H.265. In its operation, the video coder (203) may perform variouscompression operations, including predictive coding operations thatexploit temporal and spatial redundancies in the input video sequence.The coded video data, therefore, may conform to a syntax specified bythe video coding technology or standard being used.

In an embodiment, the transmitter (440) may transmit additional datawith the encoded video. The video coder (430) may include such data aspart of the coded video sequence. Additional data may comprisetemporal/spatial/SNR enhancement layers, other forms of redundant datasuch as redundant pictures and slices, Supplementary EnhancementInformation (SEI) messages, Visual Usability Information (VUI) parameterset fragments, and so on.

As discussed above, the coding of the coded video sequence can be inaccordance with a video coding technology or standard, and can followprinciples well known to a person skilled in the art, including variablelength coding, Huffman coding, arithmetic coding with or without contextsensitivity, and so forth. In some streaming systems, the videobitstreams (204, 207, 209) can be encoded according to certain videocoding/compression standards. An example of those standards includeITU-T HEVC Recommendation of H.265.

In HEVC, the primary transforms may have 4-point, 8-point, 16-point or32-point DCT-2 transform core matrices, and the transform core matricesmay be represented using 8-bit integers, (i.e., an 8-bit transform corematrix). The HEVC transform core matrices of smaller DCT-2 are part oflarger DCT-2, as shown below:

4 × 4 DCT−2 transform {64, 64, 64, 64} {83, 36, −36, −83} {64, −64, −64,64} {36, −83, 83, −36} 8 × 8 DCT−2 transform {64, 64, 64, 64, 64, 64,64, 64} {89, 75, 50, 18, −18, −50, −75, −89} {83, 36, −36, −83, −83,−36, 36, 83} {75, −18, −89, −50, 50, 89, 18, −75} {64, −64, −64, 64, 64,−64, −64, 64} {50, −89, 18, 75, −75, −18, 89, −50} {36, −83, 83, −36,−36, 83, −83, 36} {18, −50, 75, −89, 89, −75, 50, −18} 16 × 16 DCT−2transform {64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64} {90 87 80 7057 43 25 9 −9 −25 −43 −57 −70 −80 −87 −90} {89 75 50 18 −18 −50 −75 −89−89 −75 −50 −18 18 50 75 89} {87 57 9 −43 −80 −90 −70 −25 25 70 90 80 43−9 −57 −87} {83 36 −36 −83 −83 −36 36 83 83 36 −36 −83 −83 −36 36 83}{80 9 −70 −87 −25 57 90 43 −43 −90 −57 25 87 70 −9 −80} {75 −18 −89 −5050 89 18 −75 −75 18 89 50 −50 −89 −18 75} {70 −43 −87 9 90 25 −80 −57 5780 −25 −90 −9 87 43 −70} {64 −64 −64 64 64 −64 −64 64 64 −64 −64 64 64−64 −64 64} {57 −80 −25 90 −9 −87 43 70 −70 −43 87 9 −90 25 80 −57} {50−89 18 75 −75 −18 89 −50 −50 89 −18 −75 75 18 −89 50} {43 −90 57 25 −8770 9 −80 80 −9 −70 87 −25 −57 90 −43} {36 −83 83 −36 −36 83 −83 36 36−83 83 −36 −36 83 −83 36} {25 −70 90 −80 43 9 −57 87 −87 57 −9 −43 80−90 70 −25} {18 −50 75 −89 89 −75 50 −18 −18 50 −75 89 −89 75 −50 18} {9−25 43 −57 70 −80 87 −90 90 −87 80 −70 57 −43 25 −9} 32 × 32 DCT−2transform {64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 6464 64 64 64 64 64 64 64 64 64 64 64} {90 90 88 85 82 78 73 67 61 54 4638 31 22 13 4 −4 −13 −22 −31 −38 −46 −54 −61 −67 −73 −78 −82 −85 −88 −90−90} {90 87 80 70 57 43 25 9 −9 −25 −43 −57 −70 −80 −87 −90 −90 −87 −80−70 −57 −43 −25 −9 9 25 43 57 70 80 87 90} {90 82 67 46 22 −4 −31 −54−73 −85 −90 −88 −78 −61 −38 −13 13 38 61 78 88 90 85 73 54 31 4 −22 −46−67 −82 −90} {89 75 50 18 −18 −50 −75 −89 −89 −75 −50 −18 18 50 75 89 8975 50 18 −18 −50 −75 −89 −89 −75 −50 −18 18 50 75 89} {88 67 31 −13 −54−82 −90 −78 −46 −4 38 73 90 85 61 22 −22 −61 −85 −90 −73 −38 4 46 78 9082 54 13 −31 −67 −88} {87 57 9 −43 −80 −90 −70 −25 25 70 90 80 43 −9 −57−87 −87 −57 −9 43 80 90 70 25 −25 −70 −90 −80 −43 9 57 87} {85 46 −13−67 −90 −73 −22 38 82 88 54 −4 −61 −90 −78 −31 31 78 90 61 4 −54 −88 −82−38 22 73 90 67 13 −46 −85} {83 36 −36 −83 −83 −36 36 83 83 36 −36 −83−83 −36 36 83 83 36 −36 −83 −83 −36 36 83 83 36 −36 −83 −83 −36 36 83}{82 22 −54 −90 −61 13 78 85 31 −46 −90 −67 4 73 88 38 −38 −88 −73 −4 6790 46 −31 −85 −78 −13 61 90 54 −22 −82} {80 9 −70 −87 −25 57 90 43 −43−90 −57 25 87 70 −9 −80 −80 −9 70 87 25 −57 −90 −43 43 90 57 −25 −87 −709 80} {78 −4 −82 −73 13 85 67 −22 −88 −61 31 90 54 −38 −90 −46 46 90 38−54 −90 −31 61 88 22 −67 −85 −13 73 82 4 −78} {75 −18 −89 −50 50 89 18−75 −75 18 89 50 −50 −89 −18 75 75 −18 −89 −50 50 89 18 −75 −75 18 89 50−50 −89 −18 75} {73 −31 −90 −22 78 67 −38 −90 −13 82 61 −46 −88 −4 85 54−54 −85 4 88 46 −61 −82 13 90 38 −67 −78 22 90 31 −73} {70 −43 −87 9 9025 −80 −57 57 80 −25 −90 −9 87 43 −70 −70 43 87 −9 −90 −25 80 57 −57 −8025 90 9 −87 −43 70} {67 −54 −78 38 85 −22 −90 4 90 13 −88 −31 82 46 −73−61 61 73 −46 −82 31 88 −13 −90 −4 90 22 −85 −38 78 54 −67} {64 −64 −6464 64 −64 −64 64 64 −64 −64 64 64 −64 −64 64 64 −64 −64 64 64 −64 −64 6464 −64 −64 64 64 −64 −64 64} {61 −73 −46 82 31 −88 −13 90 −4 −90 22 85−38 −78 54 67 −67 −54 78 38 −85 −22 90 4 −90 13 88 −31 −82 46 73 −61}{57 −80 −25 90 −9 −87 43 70 −70 −43 87 9 −90 25 80 −57 −57 80 25 −90 987 −43 −70 70 43 −87 −9 90 −25 −80 57} {54 −85 −4 88 −46 −61 82 13 −9038 67 −78 −22 90 −31 −73 73 31 −90 22 78 −67 −38 90 −13 −82 61 46 −88 485 −54} {50 −89 18 75 −75 −18 89 −50 −50 89 −18 −75 75 18 −89 50 50 −8918 75 −75 −18 89 −50 −50 89 −18 −75 75 18 −89 50} {46 −90 38 54 −90 3161 −88 22 67 −85 13 73 −82 4 78 −78 −4 82 −73 −13 85 −67 −22 88 −61 −3190 −54 −38 90 −46} {43 −90 57 25 −87 70 9 −80 80 −9 −70 87 −25 −57 90−43 −43 90 −57 −25 87 −70 −9 80 −80 9 70 −87 25 57 −90 43} {38 −88 73 −4−67 90 −46 −31 85 −78 13 61 −90 54 22 −82 82 −22 −54 90 −61 −13 78 −8531 46 −90 67 4 −73 88 −38} {36 −83 83 −36 −36 83 −83 36 36 −83 83 −36−36 83 −83 36 36 −83 83 −36 −36 83 −83 36 36 −83 83 −36 −36 83 −83 36}{31 −78 90 −61 4 54 −88 82 −38 −22 73 −90 67 −13 −46 85 −85 46 13 −67 90−73 22 38 −82 88 −54 −4 61 −90 78 −31} {25 −70 90 −80 43 9 −57 87 −87 57−9 −43 80 −90 70 −25 −25 70 −90 80 −43 −9 57 −87 87 −57 9 43 −80 90 −7025} {22 −61 85 −90 73 −38 −4 46 −78 90 −82 54 −13 −31 67 −88 88 −67 3113 −54 82 −90 78 −46 4 38 −73 90 −85 61 −22} {18 −50 75 −89 89 −75 50−18 −18 50 −75 89 −89 75 −50 18 18 −50 75 −89 89 −75 50 −18 −18 50 −7589 −89 75 −50 18} {13 −38 61 −78 88 −90 85 −73 54 −31 4 22 −46 67 −82 90−90 82 −67 46 −22 −4 31 −54 73 −85 90 −88 78 −61 38 −13} {9 −25 43 −5770 −80 87 −90 90 −87 80 −70 57 −43 25 −9 −9 25 −43 57 −70 80 −87 90 −9087 −80 70 −57 43 −25 9} {4 −13 22 −31 38 −46 54 −61 67 −73 78 −82 85 −8890 −90 90 −90 88 −85 82 −78 73 −67 61 −54 46 −38 31 −22 13 −4}

As shown above, each of the DCT-2 cores show symmetry/anti-symmetrycharacteristics. Thus, a so-called “partial butterfly” implementationmay be supported to reduce the number of operation counts(multiplications, adds/subs, shifts), and identical results of matrixmultiplication can be obtained using a partial butterfly implementation.More details regarding HEVC transform design can be found in Non-PatentLiterature 1.

Under development is a video coding standard informally known asVersatile Video Coding (VVC). The disclosed subject matter may be usedin the context of VVC.

For a primary transform in VVC, besides 4-point, 8-point, 16-point and32-point DCT-2 transforms, which are the same as with HEVC, additional2-point, 64-point and 128-point DCT-2 may also be included. The 2-pointDCT-2 core can be an 8-bit integer matrix which is aligned with the HEVCprimary transform design, but the 64-point and 128-point DCT-2 use10-bit transform cores, which means the 2-point, 4-point, 8-point,16-point and 32-point DCT-2 transforms cannot be extracted from the64-point DCT-2 core or 128-point DCT-2 core in VVC.

Parts of an exemplary 64-point DCT-2 core defined in VVC are shownbelow, where {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an,ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf,bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw, bx,by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck}={256, 334, 139, 355,301, 201, 71, 360, 346, 319, 280, 230, 171, 105, 35, 362, 358, 351, 341,327, 311, 291, 268, 243, 216, 186, 155, 122, 88, 53, 18, 362, 361, 359,357, 353, 349, 344, 338, 331, 323, 315, 306, 296, 285, 274, 262, 250,236, 223, 208, 194, 178, 163, 147, 130, 114, 97, 79, 62, 44, 27, 9}:

{aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa}

{bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw,bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, -ck, -cj, -ci,-ch, -cg, -cf, -ce, -cd, -cc, -cb, -ca, -bz, -by, -bx, -bw, -bv, -bu,-bt, -bs, -br, -bq, -bp, -bo, -bn, -bm, -bl, -bk, -bj, -bi, -bh, -bg,-bf}{ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, -be,-bd, -bc, -bb, -ba, -az, -ay, -ax, -aw, -av, -au, -at, -as, -ar, -aq,-ap, -ap, -aq, -ar, -as, -at, -au, -av, -aw, -ax, -ay, -az, -ba, -bb,-bc, -bd, -be, be, bd, bc, bb, ba, az, ay, ax, aw, av, au, at, as, ar,aq, ap}{bg, bj, bm, bp, bs, bv, by, cb, ce, ch, ck, -ci, -cf, -cc, -bz, -bw,-bt, -bq, -bn, -bk, -bh, -bf, -bi, -bl, -bo, -br, -bu, -bx, -ca, -cd,-cg, -cj, cj, cg, cd, ca, bx, bu, br, bo, bl, bi, bf, bh, bk, bn, bq,bt, bw, bz, cc, cf, ci, -ck, -ch, -ce, -cb, -by, -bv, -bs, -bp, -bm,-bj, -bg}{ah, ai, aj, ak, al, am, an, ao, -ao, -an, -am, -al, -ak, -aj, -ai, -ah,-ah, -ai, -aj, -ak, -al, -am, -an, -ao, ao, an, am, al, ak, aj, ai, ah,ah, ai, aj, ak, al, am, an, ao, -ao, -an, -am, -al, -ak, -aj, -ai, -ah,-ah, -ai, -aj, -ak, -al, -am, -an, -ao, ao, an, am, al, ak, aj, ai, ah}{bh, bm, br, bw, cb, cg, -ck, -cf, -ca, -bv, -bq, -bl, -bg, -bi, -bn,-bs, -bx, -cc, -ch, cj, ce, bz, bu, bp, bk, bf, bj, bo, bt, by, cd, ci,-ci, -cd, -by, -bt, -bo, -bj, -bf, -bk, -bp, -bu, -bz, -ce, -cj, ch, cc,bx, bs, bn, bi, bg, bl, bq, bv, ca, cf, ck, -cg, -cb, -bw, -br, -bm,-bh}{aq, at, aw, az, bc, -be, -bb, -ay, -av, -as, -ap, -ar, -au, -ax, -ba,-bd, bd, ba, ax, au, ar, ap, as, av, ay, bb, be, -bc, -az, -aw, -at,-aq, -aq, -at, -aw, -az, -bc, be, bb, ay, av, as, ap, ar, au, ax, ba,bd, -bd, -ba, -ax, -au, -ar, -ap, -as, -av, -ay, -bb, -be, bc, az, aw,at, aq}{bi, bp, bw, cd, ck, -ce, -bx, -bq, -bj, -bh, -bo, -bv, -cc, -cj, cf,by, br, bk, bg, bn, bu, cb, ci, -cg, -bz, -bs, -bl, -bf, -bm, -bt, -ca,-ch, ch, ca, bt, bm, bf, bl, bs, bz, cg, -ci, -cb, -bu, -bn, -bg, -bk,-br, -by, -cf, cj, cc, bv, bo, bh, bj, bq, bx, ce, -ck, -cd, -bw, -bp,-bi}{ad, ae, af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad,ad, ae, af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad,ad, ae, af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad,ad, ae, af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad}{bj, bs, cb, ck, -cc, -bt, -bk, -bi, -br, -ca, -cj, cd, bu, bl, bh, bq,bz, ci, -ce, -bv, -bm, -bg, -bp, -by, -ch, cf, bw, bn, bf, bo, bx, cg,-cg, -bx, -bo, -bf, -bn, -bw, -cf, ch, bv, bp, bg, bm, bv, ce, -ci, -bz,-bq, -bh, -bl, -bu, -cd, cj, ca, br, bi, bk, bt, cc, -ck, -cb, -bs, -bj}{ar, aw, bb, -bd, -ay, -at, -ap, -au, -az, -be, ba, av, aq, as, ax, bc,-bc, -ax, -as, -aq, -av, -ba, be, az, au, ap, at, ay, bd, -bb, -aw, -ar,-ar, -aw, -bb, bd, ay, at, ap, au, az, be, -ba, -av, -aq, -as, -ax, -bc,bc, ax, as, aq, av, ba, -be, -az, -au, -ap, -at, -ay, -bd, bb, aw, ar}{bk, bv, cg, -ce, -bt, -bi, -bm, -bx, -ci, cc, br, bg, bo, bz, ck, -ca,-bp, -bf, -bq, -cb, cj, by, bn, bh, bs, cd, -ch, -bw, -bl, -bj, -bu,-cf, cf, bu, bj, bl, bw, ch, -cd, -bs, -bh, -bn, -by, -cj, cb, bq, bf,bp, ca, -ck, -bz, -bo, -bg, -br, -cc, ci, bx, bm, bi, bt, ce, -cg, -bv,-bk}{ai, al, ao, -am, -aj, -ah, -ak, -an, an, ak, ah, aj, am, -ao, -al, -ai,-ai, -al, -ao, am, aj, ah, ak, an, -an, -ak, -ah, -aj, -am, ao, al, ai,ai, al, ao, -am, -aj, -ah, -ak, -an, an, ak, ah, aj, am, -ao, -al, -ai,-ai, -al, -ao, am, aj, ah, ak, an, -an, -ak, -ah, -aj, -am, ao, al, ai}{bl, by, -ck, -bx, -bk, -bm, -bz, cj, bw, bj, bn, ca, -ci, -bv, -bi,-bo, -cb, ch, bu, bh, bp, cc, -cg, -bt, -bg, -bq, -cd, cf, bs, bf, br,ce, -ce, -br, -bf, -bs, -cf, cd, bq, bg, bt, cg, -cc, -bp, -bh, -bu,-ch, cb, bo, bi, bv, ci, -ca, -bn, -bj, -bw, -cj, bz, bm, bk, bx, ck,-by, -bl}{as, az, -bd, -aw, -ap, -av, -bc, ba, at, ar, ay, -be, -ax, -aq, -au,-bb, bb, au, aq, ax, be, -ay, -ar, -at, -ba, bc, av, ap, aw, bd, -az,-as, -as, -az, bd, aw, ap, av, bc, -ba, -at, -ar, -ay, be, ax, aq, au,bb, -bb, -au, -aq, -ax, -be, ay, ar, at, ba, -bc, -av, -ap, -aw, -bd,az, as}{bm, cb, -cf, -bq, -bi, -bx, cj, bu, bf, bt, ci, -by, -bj, -bp, -ce, cc,bn, bl, ca, -cg, -br, -bh, -bw, ck, bv, bg, bs, ch, -bz, -bk, -bo, -cd,cd, bo, bk, bz, -ch, -bs, -bg, -bv, -ck, bw, bh, br, cg, -ca, -bl, -bn,-cc, ce, bp, bj, by, -ci, -bt, -bf, -bu, -cj, bx, bi, bq, cf, -cb, -bm}{ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab,ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab,ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab,ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab}{bn, ce, -ca, -bj, -br, -ci, bw, bf, bv, -cj, -bs, -bi, -bz, cf, bo, bm,cd, -cb, -bk, -bq, -ch, bx, bg, bu, -ck, -bt, -bh, -by, cg, bp, bl, cc,-cc, -bl, -bp, -cg, by, bh, bt, ck, -bu, -bg, -bx, ch, bq, bk, cb, -cd,-bm, -bo, -cf, bz, bi, bs, cj, -bv, -bf, -bw, ci, br, bj, ca, -ce, -bn}{at, bc, -ay, -ap, -ax, bd, au, as, bb, -az, -aq, -aw, be, av, ar, ba,-ba, -ar, -av, -be, aw, aq, az, -bb, -as, -au, -bd, ax, ap, ay, -bc,-at, -at, -bc, ay, ap, ax, -bd, -au, -as, -bb, az, aq, aw, -be, -av,-ar, -ba, ba, ar, av, be, -aw, -aq, -az, bb, as, au, bd, -ax, -ap, -ay,bc, at}{bo, ch, -bv, -bh, -ca, cc, bj, bt, -cj, -bq, -bm, -cf, bx, bf, by, -ce,-bl, -br, -ck, bs, bk, cd, -bz, -bg, -bw, cg, bn, bp, ci, -bu, -bi, -cb,cb, bi, bu, -ci, -bp, -bn, -cg, bw, bg, bz, -cd, -bk, -bs, ck, br, bl,ce, -by, -bf, -bx, cf, bm, bq, cj, -bt, -bj, -cc, ca, bh, bv, -ch, -bo}{aj, ao, -ak, -ai, -an, al, ah, am, -am, -ah, -al, an, ai, ak, -ao, -aj,-aj, -ao, ak, ai, an, -al, -ah, -am, am, ah, al, -an, -ai, -ak, ao, aj,aj, ao, -ak, -ai, -an, al, ah, am, -am, -ah, -al, an, ai, ak, -ao, -aj,-aj, -ao, ak, ai, an, -al, -ah, -am, am, ah, al, -an, -ai, -ak, ao, aj}{bp, ck, -bq, -bo, -cj, br, bn, ci, -bs, -bm, -ch, bt, bl, cg, -bu, -bk,-cf, bv, bj, ce, -bw, -bi, -cd, bx, bh, cc, -by, -bg, -cb, bz, bf, ca,-ca, -bf, -bz, cb, bg, by, -cc, -bh, -bx, cd, bi, bw, -ce, -bj, -bv, cf,bk, bu, -cg, -bl, -bt, ch, bm, bs, -ci, -bn, -br, cj, bo, bq, -ck, -bp}{au, -be, -at, -av, bd, as, aw, -bc, -ar, -ax, bb, aq, ay, -ba, -ap,-az, az, ap, ba, -ay, -aq, -bb, ax, ar, bc, -aw, -as, -bd, av, at, be,-au, -au, be, at, av, -bd, -as, -aw, bc, ar, ax, -bb, -aq, -ay, ba, ap,az, -az, -ap, -ba, ay, aq, bb, -ax, -ar, -bc, aw, as, bd, -av, -at, -be,au}{bq, -ci, -bl, -bv, cd, bg, ca, -by, -bi, -cf, bt, bn, ck, -bo, -bs, cg,bj, bx, -cb, -bf, -cc, bw, bk, ch, -br, -bp, cj, bm, bu, -ce, -bh, -bz,bz, bh, ce, -bu, -bm, -cj, bp, br, -ch, -bk, -bw, cc, bf, cb, -bx, -bj,-cg, bs, bo, -ck, -bn, -bt, cf, bi, by, -ca, -bg, -cd, bv, bl, ci, -bq}{ae, -ag, -ad, -af, af, ad, ag, -ae, -ae, ag, ad, af, -af, -ad, -ag, ae,ae, -ag, -ad, -af, af, ad, ag, -ae, -ae, ag, ad, af, -af, -ad, -ag, ae,ae, -ag, -ad, -af, af, ad, ag, -ae, -ae, ag, ad, af, -af, -ad, -ag, ae,ae, -ag, -ad, -af, af, ad, ag, -ae, -ae, ag, ad, af, -af, -ad, -ag, ae}{br, -cf, -bg, -cc, bu, bo, -ci, -bj, -bz, bx, bl, ck, -bm, -bw, ca, bi,ch, -bp, -bt, cd, bf, ce, -bs, -bq, cg, bh, cb, -bv, -bn, cj, bk, by,-by, -bk, -cj, bn, bv, -cb, -bh, -cg, bq, bs, -ce, -bf, -cd, bt, bp,-ch, -bi, -ca, bw, bm, -ck, -bl, -bx, bz, bj, ci, -bo, -bu, cc, bg, cf,-br}{av, -bb, -ap, -bc, au, aw, -ba, -aq, -bd, at, ax, -az, -ar, -be, as,ay, -ay, -as, be, ar, az, -ax, -at, bd, aq, ba, -aw, -au, bc, ap, bb,-av, -av, bb, ap, bc, -au, -aw, ba, aq, bd, -at, -ax, az, ar, be, -as,-ay, ay, as, -be, -ar, -az, ax, at, -bd, -aq, -ba, aw, au, -bc, -ap,-bb, av}{bs, -cc, -bi, -cj, bl, bz, -bv, -bp, cf, bf, cg, -bo, -bw, by, bm, -ci,-bh, -cd, br, bt, -cb, -bj, -ck, bk, ca, -bu, -bq, ce, bg, ch, -bn, -bx,bx, bn, -ch, -bg, -ce, bq, bu, -ca, -bk, ck, bj, cb, -bt, -br, cd, bh,ci, -bm, -by, bw, bo, -cg, -bf, -cf, bp, bv, -bz, -bl, cj, bi, cc, -bs}{ak, -am, -ai, ao, ah, an, -aj, -al, al, aj, -an, -ah, -ao, ai, am, -ak,-ak, am, ai, -ao, -ah, -an, aj, al, -al, -aj, an, ah, ao, -ai, -am, ak,ak, -am, -ai, ao, ah, an, -aj, -al, al, aj, -an, -ah, -ao, ai, am, -ak,-ak, am, ai, -ao, -ah, -an, aj, al, -al, -aj, an, ah, ao, -ai, -am, ak}{bt, -bz, -bn, cf, bh, ck, -bi, -ce, bo, bv, -bu, -bs, ca, bm, -cg, -bg,-cj, bj, cd, -bp, -bx, bv, br, -cb, -bl, ch, bf, ci, -bk, -cc, bq, bw,-bw, -bq, cc, bk, -ci, -bf, -ch, bl, cb, -br, -bv, bx, bp, -cd, -bj, cj,bg, cg, -bm, -ca, bs, bu, -by, -bo, ce, bi, -ck, -bh, -cf, bn, bz, -bt}{aw, -ay, -au, ba, as, -bc, -aq, be, ap, bd, -ar, -bb, at, az, -av, -ax,ax, av, -az, -at, bb, ar, -bd, -ap, -be, aq, bc, -as, -ba, au, ay, -aw,-aw, ay, au, -ba, -as, bc, aq, -be, -ap, -bd, ar, bb, -at, -az, av, ax,-ax, -av, az, at, -bb, -ar, bd, ap, be, -aq, -bc, as, ba, -au, -ay, aw}{bu, -bw, -bs, by, bq, -ca, -bo, cc, bm, -ce, -bk, cg, bi, -ci, -bg, ck,bf, cj, -bh, -ch, bj, cf, -bl, -cd, bn, cb, -bp, -bz, br, bx, -bt, -bv,by, bt, -bx, -br, bz, bp, -cb, -bn, cd, bl, -cf, -bj, ch, bh, -cj, -bf,-ck, bg, ci, -bi, -cg, bk, ce, -bm, -cc, bo, ca, -bq, -by, bs, bw, -bu}{aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa,aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa,aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa,aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa, aa, -aa, -aa, aa}{bv, -bt, -bx, br, bz, -bp, -cb, bn, cd, -bl, -cf, bj, ch, -bh, -cj, bf,-ck, -bg, ci, bi, -cg, -bk, ce, bm, -cc, -bo, ca, bq, -by, -bs, bw, bu,-bu, -bw, bs, by, -bq, -ca, bo, cc, -bm, -ce, bk, cg, -bi, -ci, bg, ck,-bf, cj, bh, -ch, -bj, cf, bl, -cd, -bn, cb, bp, -bz, -br, bx, bt, -by}{ax, -av, -az, at, bb, -ar, -bd, ap, -be, -aq, bc, as, -ba, -au, ay, aw,-aw, -ay, au, ba, -as, -bc, aq, be, -ap, bd, ar, -bb, -at, az, av, -ax,-ax, av, az, -at, -bb, ar, bd, -ap, be, aq, -bc, -as, ba, au, -ay, -aw,aw, ay, -au, -ba, as, bc, -aq, -be, ap, -bd, -ar, bb, at, -az, -av, ax}{bw, -bq, -cc, bk, ci, -bf, ch, bl, -cb, -br, bv, bx, -bp, -cd, bj, cj,-bg, cg, bm, -ca, -bs, bu, by, -bo, -ce, bi, ck, -bh, cf, bn, -bz, -bt,bt, bz, -bn, -cf, bh, -ck, -bi, ce, bo, -bv, -bu, bs, ca, -bm, -cg, bg,-cj, -bj, cd, bp, -bx, -bv, br, cb, -bl, -ch, bf, -ci, -bk, cc, bq, -bw}{al, -aj, -an, ah, -ao, -ai, am, ak, -ak, -am, ai, ao, -ah, an, aj, -al,-al, aj, an, -ah, ao, ai, -am, -ak, ak, am, -ai, -ao, ah, -an, -aj, al,al, -aj, -an, ah, -ao, -ai, am, ak, -ak, -am, ai, ao, -ah, an, aj, -al,-al, aj, an, -ah, ao, ai, -am, -ak, ak, am, -ai, -ao, ah, -an, -aj, al}{bx, -bn, -ch, bg, -ce, -bq, bu, ca, -bk, -ck, bj, -cb, -bt, br, cd,-bh, ci, bm, -by, -bw, bo, cg, -bf, cf, bp, -bv, -bz, bl, cj, -bi, cc,bs, -bs, -cc, bi, -cj, -bl, bz, bv, -bp, -cf, bf, -cg, -bo, bw, by, -bm,-ci, bh, -cd, -br, bt, cb, -bj, ck, bk, -ca, -bu, bq, ce, -bg, ch, bn,-bx}{ay, -as, -be, ar, -az, -ax, at, bd, -aq, ba, aw, -au, -bc, ap, -bb,-av, av, bb, -ap, bc, au, -aw, -ba, aq, -bd, -at, ax, az, -ar, be, as,-ay, -ay, as, be, -ar, az, ax, -at, -bd, aq, -ba, -aw, au, bc, -ap, bb,av, -av, -bb, ap, -bc, -au, aw, ba, -aq, bd, at, -ax, -az, ar, -be, -as,ay}{by, -bk, cj, bn, -bv, -cb, bh, -cg, -bq, bs, ce, -bf, cd, bt, -bp, -ch,bi, -ca, -bw, bm, ck, -bl, bx, bz, -bj, ci, bo, -bu, -cc, bg, -cf, -br,br, cf, -bg, cc, bu, -bo, -ci, bj, -bz, -bx, bl, -ck, -bm, bw, ca, -bi,ch, bp, -bt, -cd, bf, -ce, -bs, bq, cg, -bh, cb, bv, -bn, -cj, bk, -by}{af, -ad, ag, ae, -ae, -ag, ad, -af, -af, ad, -ag, -ae, ae, ag, -ad, af,af, -ad, ag, ae, -ae, -ag, ad, -af, -af, ad, -ag, -ae, ae, ag, -ad, af,af, -ad, ag, ae, -ae, -ag, ad, -af, -af, ad, -ag, -ae, ae, ag, -ad, af,af, -ad, ag, ae, -ae, -ag, ad, -af, -af, ad, -ag, -ae, ae, ag, -ad, af}{bz, -bh, ce, bu, -bm, cj, bp, -br, -ch, bk, -bw, -cc, bf, -cb, -bx, bj,-cg, -bs, bo, ck, -bn, bt, cf, -bi, by, ca, -bg, cd, bv, -bl, ci, bq,-bq, -ci, bl, -bv, -cd, bg, -ca, -by, bi, -cf, -bt, bn, -ck, -bo, bs,cg, -bj, bx, cb, -bf, cc, bw, -bk, ch, br, -bp, -cj, bm, -bu, -ce, bh,-bz}{az, -ap, ba, ay, -aq, bb, ax, -ar, bc, aw, -as, bd, av, -at, be, au,-au, -be, at, -av, -bd, as, -aw, -bc, ar, -ax, -bb, aq, -ay, -ba, ap,-az, -az, ap, -ba, -ay, aq, -bb, -ax, ar, -bc, -aw, as, -bd, -av, at,-be, -au, au, be, -at, av, bd, -as, aw, bc, -ar, ax, bb, -aq, ay, ba,-ap, az}{ca, -bf, bz, cb, -bg, by, cc, -bh, bx, cd, -bi, bw, ce, -bj, bv, cf,-bk, bu, cg, -bl, bt, ch, -bm, bs, ci, -bn, br, cj, -bo, bq, ck, -bp,bp, -ck, -bq, bo, -cj, -br, bn, -ci, -bs, bm, -ch, -bt, bl, -cg, -bu,bk, -cf, -bv, bj, -ce, -bw, bi, -cd, -bx, bh, -cc, -by, bg, -cb, -bz,bf, -ca}{am, -ah, al, an, -ai, ak, ao, -aj, aj, -ao, -ak, ai, -an, -al, ah, -am,-am, ah, -al, -an, ai, -ak, -ao, aj, -aj, ao, ak, -ai, an, al, -ah, am,am, -ah, al, an, -ai, ak, ao, -aj, aj, -ao, -ak, ai, -an, -al, ah, -am,-am, ah, -al, -an, ai, -ak, -ao, aj, -aj, ao, ak, -ai, an, al, -ah, am}{cb, -bi, bu, ci, -bp, bn, -cg, -bw, bg, -bz, -cd, bk, -bs, -ck, br,-bl, ce, by, -bf, bx, cf, -bm, bq, -cj, -bt, bj, -cc, -ca, bh, -bv, -ch,bo, -bo, ch, bv, -bh, ca, cc, -bj, bt, cj, -bq, bm, -cf, -bx, bf, -by,-ce, bl, -br, ck, bs, -bk, cd, bz, -bg, bw, cg, -bn, bp, -ci, -bu, bi,-cb}{ba, -ar, av, -be, -aw, aq, -az, -bb, as, -au, bd, ax, -ap, ay, bc, -at,at, -bc, -ay, ap, -ax, -bd, au, -as, bb, az, -aq, aw, be, -av, ar, -ba,-ba, ar, -av, be, aw, -aq, az, bb, -as, au, -bd, -ax, ap, -ay, -bc, at,-at, bc, ay, -ap, ax, bd, -au, as, -bb, -az, aq, -aw, -be, av, -ar, ba}{cc, -bl, bp, -cg, -by, bh, -bt, ck, bu, -bg, bx, ch, -bq, bk, -cb, -cd,bm, -bo, cf, bz, -bi, bs, -cj, -bv, bf, -bw, -ci, br, -bj, ca, ce, -bn,bn, -ce, -ca, bj, -br, ci, bw, -bf, bv, cj, -bs, bi, -bz, -cf, bo, -bm,cd, cb, -bk, bq, -ch, -bx, bg, -bu, -ck, bt, -bh, by, cg, -bp, bl, -cc}{ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac,ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac,ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac,ac, -ab, ab, -ac, -ac, ab, -ab, ac, ac, -ab, ab, -ac, -ac, ab, -ab, ac}{cd, -bo, bk, -bz, -ch, bs, -bg, bv, -ck, -bw, bh, -br, cg, ca, -bl, bn,-cc, -ce, bp, -bj, by, ci, -bt, bf, -bu, cj, bx, -bi, bq, -cf, -cb, bm,-bm, cb, cf, -bq, bi, -bx, -cj, bu, -bf, bt, -ci, -by, bj, -bp, ce, cc,-bn, bl, -ca, -cg, br, -bh, bw, ck, -bv, bg, -bs, ch, bz, -bk, bo, -cd}{bb, -au, aq, -ax, be, ay, -ar, at, -ba, -bc, av, -ap, aw, -bd, -az, as,-as, az, bd, -aw, ap, -av, bc, ba, -at, ar, -ay, -be, ax, -aq, au, -bb,-bb, au, -aq, ax, -be, -ay, ar, -at, ba, bc, -av, ap, -aw, bd, az, -as,as, -az, -bd, aw, -ap, av, -bc, -ba, at, -ar, ay, be, -ax, aq, -au, bb}{ce, -br, bf, -bs, cf, cd, -bq, bg, -bt, cg, cc, -bp, bh, -bu, ch, cb,-bo, bi, -bv, ci, ca, -bn, bj, -bw, cj, bz, -bm, bk, -bx, ck, by, -bl,bl, -by, -ck, bx, -bk, bm, -bz, -cj, bw, -bj, bn, -ca, -ci, bv, -bi, bo,-cb, -ch, bu, -bh, bp, -cc, -cg, bt, -bg, bq, -cd, -cf, bs, -bf, br,-ce}{an, -ak, ah, -aj, am, ao, -al, ai, -ai, al, -ao, -am, aj, -ah, ak, -an,-an, ak, -ah, aj, -am, -ao, al, -ai, ai, -al, ao, am, -aj, ah, -ak, an,an, -ak, ah, -aj, am, ao, -al, ai, -ai, al, -ao, -am, aj, -ah, ak, -an,-an, ak, -ah, aj, -am, -ao, al, -ai, ai, -al, ao, am, -aj, ah, -ak, an}{cf, -bu, bj, -bl, bw, -ch, -cd, bs, -bh, bn, -by, cj, cb, -bq, bf, -bp,ca, ck, -bz, bo, -bg, br, -cc, -ci, bx, -bm, bi, -bt, ce, cg, -bv, bk,-bk, bv, -cg, -ce, bt, -bi, bm, -bx, ci, cc, -br, bg, -bo, bz, -ck, -ca,bp, -bf, bq, -cb, -cj, by, -bn, bh, -bs, cd, ch, -bw, bl, -bj, bu, -cf}{bc, -ax, as, -aq, av, -ba, -be, az, -au, ap, -at, ay, -bd, -bb, aw,-ar, ar, -aw, bb, bd, -ay, at, -ap, au, -az, be, ba, -av, aq, -as, ax,-bc, -bc, ax, -as, aq, -av, ba, be, -az, au, -ap, at, -ay, bd, bb, -aw,ar, -ar, aw, -bb, -bd, ay, -at, ap, -au, az, -be, -ba, av, -aq, as, -ax,be}{cg, -bx, bo, -bf, bn, -bw, cf, ch, -by, bp, -bg, bm, -bv, ce, ci, -bz,bq, -bh, bl, -bu, cd, cj, -ca, br, -bi, bk, -bt, cc, ck, -cb, bs, -bj,bj, -bs, cb, -ck, -cc, bt, -bk, bi, -br, ca, -cj, -cd, bu, -bl, bh, -bq,bz, -ci, -ce, bv, -bm, bg, -bp, by, -ch, -cf, bw, -bn, bf, -bo, bx, -cg}{ag, -af, ae, -ad, ad, -ae, af, -ag, -ag, af, -ae, ad, -ad, ae, -af, ag,ag, -af, ae, -ad, ad, -ae, af, -ag, -ag, af, -ae, ad, -ad, ae, -af, ag,ag, -af, ae, -ad, ad, -ae, af, -ag, -ag, af, -ae, ad, -ad, ae, -af, ag,ag, -af, ae, -ad, ad, -ae, af, -ag, -ag, af, -ae, ad, -ad, ae, -af, ag}{ch, -ca, bt, -bm, bf, -bl, bs, -bz, cg, ci, -cb, bu, -bn, bg, -bk, br,-by, cf, cj, -cc, bv, -bo, bh, -bj, bq, -bx, ce, ck, -cd, bw, -bp, bi,-bi, bp, -bw, cd, -ck, -ce, bx, -bq, bj, -bh, bo, -bv, cc, -cj, -cf, by,-br, bk, -bg, bn, -bu, cb, -ci, -cg, bz, -bs, bl, -bf, bm, -bt, ca, -ch}{bd, -ba, ax, -au, ar, -ap, as, -av, ay, -bb, be, bc, -az, aw, -at, aq,-aq, at, -aw, az, -bc, -be, bb, -ay, av, -as, ap, -ar, au, -ax, ba, -bd,-bd, ba, -ax, au, -ar, ap, -as, av, -ay, bb, -be, -bc, az, -aw, at, -aq,aq, -at, aw, -az, bc, be, -bb, ay, -av, as, -ap, ar, -au, ax, -ba, bd}{ci, -cd, by, -bt, bo, -bj, bf, -bk, bp, -bu, bz, -ce, cj, ch, -cc, bx,-bs, bn, -bi, bg, -bl, bq, -bv, ca, -cf, ck, cg, -cb, bw, -br, bm, -bh,bh, -bm, br, -bw, cb, -cg, -ck, cf, -ca, bv, -bq, bl, -bg, bi, -bn, bs,-bx, cc, -ch, -cj, ce, -bz, bu, -bp, bk, -bf, bj, -bo, bt, -by, cd, -ci}{ao, -an, am, -al, ak, -aj, ai, -ah, ah, -ai, aj, -ak, al, -am, an, -ao,-ao, an, -am, al, -ak, aj, -ai, ah, -ah, ai, -aj, ak, -al, am, -an, ao,ao, -an, am, -al, ak, -aj, ai, -ah, ah, -ai, aj, -ak, al, -am, an, -ao,-ao, an, -am, al, -ak, aj, -ai, ah, -ah, ai, -aj, ak, -al, am, -an, ao}{cj, -cg, cd, -ca, bx, -bu, br, -bo, bl, -bi, bf, -bh, bk, -bn, bq, -bt,bw, -bz, cc, -cf, ci, ck, -ch, ce, -cb, by, -bv, bs, -bp, bm, -bj, bg,-bg, bj, -bm, bp, -bs, bv, -by, cb, -ce, ch, -ck, -ci, cf, -cc, bz, -bw,bt, -bq, bn, -bk, bh, -bf, bi, -bl, bo, -br, bu, -bx, ca, -cd, cg, -cj}{be, -bd, bc, -bb, ba, -az, ay, -ax, aw, -av, au, -at, as, -ar, aq, -ap,ap, -aq, ar, -as, at, -au, av, -aw, ax, -ay, az, -ba, bb, -bc, bd, -be,-be, bd, -bc, bb, -ba, az, -ay, ax, -aw, av, -au, at, -as, ar, -aq, ap,-ap, aq, -ar, as, -at, au, -av, aw, -ax, ay, -az, ba, -bb, bc, -bd, be}{ck, -cj, ci, -ch, cg, -cf, ce, -cd, cc, -cb, ca, -bz, by, -bx, bw, -bv,bu, -bt, bs, -br, bq, -bp, bo, -bn, bm, -bl, bk, -bj, bi, -bh, bg, -bf,bf, -bg, bh, -bi, bj, -bk, bl, -bm, bn, -bo, bp, -bq, br, -bs, bt, -bu,bv, -bw, bx, -by, bz, -ca, cb, -cc, cd, -ce, cf, -cg, ch, -ci, cj, -ck}

An exemplary 128-point DCT-2 core defined in VVC is shown below, where{aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar,as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj,bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb,cc, cd, ce, cf, cg, ch, ci, cj, ck, cl, cm, cn, co, cp, cq, cr, cs, ct,cu, cv, cw, cx, cy, cz, da, db, dc, dd, de, df, dg, dh, di, dj, dk, dl,dm, dn, d_, dp, dq, dr, ds, dt, du, dv, dw, dx, dy, dz, ea, eb, ec, ed,ee, ef, eg, eh, ei, ej, ek, el, em, en, eo, ep, eq, er, es, et, eu, ev,ew}={256, 334, 139, 355, 301, 201, 71, 360, 346, 319, 280, 230, 171,105, 35, 362, 358, 351, 341, 327, 311, 291, 268, 243, 216, 186, 155,122, 88, 53, 18, 362, 361, 359, 357, 353, 349, 344, 338, 331, 323, 315,306, 296, 285, 274, 262, 250, 236, 223, 208, 194, 178, 163, 147, 130,114, 97, 79, 62, 44, 27, 9, 362, 362, 361, 361, 360, 359, 357, 356, 354,352, 350, 348, 345, 342, 339, 336, 333, 329, 325, 321, 317, 313, 308,303, 299, 293, 288, 283, 277, 271, 265, 259, 253, 246, 240, 233, 226,219, 212, 205, 197, 190, 182, 175, 167, 159, 151, 143, 134, 126, 118,109, 101, 92, 84, 75, 66, 58, 49, 40, 31, 22, 13, 4}.

The 128-point DCT-2 core is shown below, only the left half of the128×128 transform core is described, the right half can be derivedaccording to the symmetry/anti-symmetry characteristics of DCT-2:

{aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa, aa,aa, aa, aa, aa, aa, aa, aa, aa, aa, aa}

{cl, cm, cn, co, cp, cq, cr, cs, ct, cu, cv, cw, cx, cy, cz, da, db, dc,dd, de, df, dg, dh, di, dj, dk, dl, dm, dn, d_, dp, dq, dr, ds, dt, du,dv, dw, dx, dy, dz, ea, eb, ec, ed, ee, ef, eg, eh, ei, ej, ek, el, em,en, eo, ep, eq, er, es, et, eu, ev, ew}

{bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw,bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck, -ck, -cj, -ci,-ch, -cg, -cf, -ce, -cd, -cc, -cb, -ca, -bz, -by, -bx, -bw, -bv, -bu,-bt, -bs, -br, -bq, -bp, -bo, -bn, -bm, -bl, -bk, -bj, -bi, -bh, -bg,-bf}{cm, cp, cs, cv, cy, db, de, dh, dk, dn, dq, dt, dw, dz, ec, ef, ei, el,eo, er, eu, -ew, -et, -eq, -en, -ek, -eh, -ee, -eb, -dy, -dv, -ds, -dp,-dm, -dj, -dg, -dd, -da, -cx, -cu, -cr, -co, -cl, -cn, -cq, -ct, -cw,-cz, -dc, -df, -di, -dl, -d_, -dr, -du, -dx, -ea, -ed, -eg, -ej, -em,-ep, -es, -ev}{ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, -be,-bd, -b c, -bb, -ba, -az, -ay, -ax, -aw, -av, -au, -at, -as, -ar, -aq,-ap, -ap, -aq, -ar, -as, -at, -au, -av, -aw, -ax, -ay, -az, -ba, -bb, -bc, -bd, -be, be, bd, bc, bb, ba, az, ay, ax, aw, av, au, at, as, ar, aq,ap}{cn, cs, cx, dc, dh, dm, dr, dw, eb, eg, el, eq, ev, -et, -eo, -ej, -ee,-dz, -du, -dp, -dk, -df, -da, -cv, -cq, -cl, -cp, -cu, -cz, -de, -dj,-d_, -dt, -dy, -ed, -ei, -en, -es, ew, er, em, eh, ec, dx, ds, dn, di,dd, cy, ct, co, cm, Cr, cw, db, dg, dl, dq, dv, ea, ef, ek, ep, eu}{bg, bj, bm, bp, bs, bv, by, cb, ce, ch, ck, -ci, -cf, -cc, -bz, -bw,-bt, -bq, -bn, -bk, -bh, -bf, -bi, -bl, -bo, -br, -bu, -bx, -ca, -cd,-cg, -cj, cj, cg, cd, ca, bx, bu, br, bo, bl, bi, bf, bh, bk, bn, bq,bt, bw, bz, cc, cf, ci, -ck, -ch, -ce, -cb, -by, -bv, -bs, -bp, -bm,-bj, -bg}{co, cv, dc, dj, dq, dx, ee, el, es, -eu, -en, -eg, -dz, -ds, -dl, -de,-cx, -cq, -cm, -ct, -da, -dh, -d_, -dv, -ec, -ej, -eq, ew, ep, ei, eb,du, dn, dg, cz, cs, cl, Cr, cy, df, dm, dt, ea, eh, eo, ev, -er, -ek,-ed, -dw, -dp, -di, -db, -cu, -cn, -cp, -cw, -dd, -dk, -dr, -dy, -ef,-em, -et}{ah, ai, aj, ak, al, am, an, ao, -ao, -an, -am, -al, -ak, -aj, -ai, -ah,-ah, -ai, -aj, -ak, -al, -am, -an, -ao, ao, an, am, al, ak, aj, ai, ah,ah, ai, aj, ak, al, am, an, ao, -ao, -an, -am, -al, -ak, -aj, -ai, -ah,-ah, -ai, -aj, -ak, -al, -am, -an, -ao, ao, an, am, al, ak, aj, ai, ah}{cp, cy, dh, dq, dz, ei, er, -et, -ek, -eb, -ds, -dj, -da, -cr, -cn,-cw, -df, -d_, -dx, -eg, -ep, ev, em, ed, du, dl, dc, ct, cl, cu, dd,dm, dv, ee, en, ew, -eo, -ef, -dw, -dn, -de, -cv, -cm, -cs, -db, -dk,-dt, -ec, -el, -eu, eq, eh, dy, dp, dg, cx, co, cq, cz, di, dr, ea, ej,es}{bh, bm, br, bw, cb, cg, -ck, -cf, -ca, -bv, -bq, -bl, -bg, -bi, -bn,-bs, -bx, -cc, -ch, cj, ce, bz, bu, bp, bk, bf, bj, bo, bt, by, cd, ci,-ci, -cd, -by, -bt, -bo, -bj, -bf, -bk, -bp, -bu, -bz, -ce, -cj, ch, cc,bx, bs, bn, bi, bg, bl, bq, bv, ca, cf, ck, -cg, -cb, -bw, -br, -bm,-bh}{cq, db, dm, dx, ei, et, -ep, -ee, -dt, -di, -cx, -cm, -cu, -df, -dq,-eb, -em, ew, el, ea, dp, de, ct, cn, cy, dj, du, ef, eq, -es, -eh, -dw,-dl, -da, -cp, -cr, -dc, -dn, -dy, -ej, -eu, eo, ed, ds, dh, cw, cl, cv,dg, dr, ec, en, -ev, -ek, -dz, -d_, -dd, -cs, -co, -cz, -dk, -dv, -eg,-er}{aq, at, aw, az, bc, -be, -bb, -ay, -av, -as, -ap, -ar, -au, -ax, -ba,-bd, bd, ba, ax, au, ar, ap, as, av, ay, bb, be, -bc, -az, -aw, -at,-aq, -aq, -at, -aw, -az, -bc, be, bb, ay, av, as, ap, ar, au, ax, ba,bd, -bd, -ba, -ax, -au, -ar, -ap, -as, -av, -ay, -bb, -be, bc, az, aw,at, aq}{cr, de, dr, ee, er, -ep, -ec, -dp, -dc, -cp, -ct, -dg, -dt, -eg, -et,en, ea, dn, da, cn, cv, di, dv, ei, ev, -el, -dy, -dl, -cy, -cl, -cx,-dk, -dx, -ek, ew, ej, dw, dj, cw, cm, cz, dm, dz, em, -eu, -eh, -du,-dh, -cu, -co, -db, -d_, -eb, -eo, es, ef, ds, df, cs, cq, dd, dq, ed,eq}{bi, bp, bw, cd, ck, -ce, -bx, -bq, -bj, -bh, -bo, -bv, -cc, -cj, cf,by, br, bk, bg, bn, bu, cb, ci, -cg, -bz, -bs, -bl, -bf, -bm, -bt, -ca,-ch, ch, ca, bt, bm, bf, bl, bs, bz, cg, -ci, -cb, -bu, -bn, -bg, -bk,-br, -by, -cf, cj, cc, bv, bo, bh, bj, bq, bx, ce, -ck, -cd, -bw, -bp,-bi}{cs, dh, dw, el, -et, -ee, -dp, -da, -cl, -cz, -d_, -ed, -es, em, dx,di, ct, Cr, dg, dv, ek, -eu, -ef, -dq, -db, -cm, -cy, -dn, -ec, -er, en,dy, dj, cu, cq, df, du, ej, -ev, -eg, -dr, -dc, -cn, -cx, -dm, -eb, -eq,eo, dz, dk, cv, cp, de, dt, ei, -ew, -eh, -ds, -dd, -co, -cw, -dl, -ea,-ep}{ad, ae, af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad,ad, ae, af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad,ad, ae, af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad,ad, ae, af, ag, -ag, -af, -ae, -ad, -ad, -ae, -af, -ag, ag, af, ae, ad}{ct, dk, eb, es, -ek, -dt, -dc, -cl, -db, -ds, -ej, et, ec, dl, cu, cs,dj, ea, er, -el, -du, -dd, -cm, -da, -dr, -ei, eu, ed, dm, cv, cr, di,dz, eq, -em, -dv, -de, -cn, -cz, -dq, -eh, ev, ee, dn, cw, cq, dh, dy,ep, -en, -dw, -df, -co, -cy, -dp, -eg, ew, ef, d_, cx, cp, dg, dx, eo}{bj, bs, cb, ck, -cc, -bt, -bk, -bi, -br, -ca, -cj, cd, bu, bl, bh, bq,bz, ci, -ce, -bv, -bm, -bg, -bp, -by, -ch, cf, bw, bn, bf, bo, bx, cg,-cg, -bx, -bo, -bf, -bn, -bw, -cf, ch, by, bp, bg, bm, bv, ce, -ci, -bz,-bq, -bh, -bl, -bu, -cd, cj, ca, br, bi, bk, bt, cc, -ck, -cb, -bs, -bj}{cu, dn, eg, -eu, -eb, -di, -cp, -cz, -ds, -el, ep, dw, dd, cl, de, dx,eq, -ek, -dr, -cy, -cq, -dj, -ec, -ev, ef, dm, ct, cv, d_, eh, -et, -ea,-dh, -co, -da, -dt, -em, eo, dv, dc, cm, df, dy, er, -ej, -dq, -cx, -cr,-dk, -ed, -ew, ee, dl, cs, cw, dp, ei, -es, -dz, -dg, -cn, -db, -du,-en}{ar, aw, bb, -bd, -ay, -at, -ap, -au, -az, -be, ba, av, aq, as, ax, bc,-bc, -ax, -as, -aq, -av, -ba, be, az, au, ap, at, ay, bd, -bb, -aw, -ar,-ar, -aw, -bb, bd, ay, at, ap, au, az, be, -ba, -av, -aq, -as, -ax, -bc,bc, ax, as, aq, av, ba, -be, -az, -au, -ap, -at, -ay, -bd, bb, aw, ar}{cv, dq, el, -en, -ds, -cx, -ct, -d_, -ej, ep, du, cz, cr, dm, eh, -er,-dw, -db, -cp, -dk, -ef, et, dy, dd, cn, di, ed, -ev, -ea, -df, -cl,-dg, -eb, -ew, ec, dh, cm, de, dz, eu, -ee, -dj, -co, -dc, -dx, -es, eg,dl, cq, da, dv, eq, -ei, -dn, -cs, -cy, -dt, -eo, ek, dp, cu, cw, dr,em}{bk, bv, cg, -ce, -bt, -bi, -bm, -bx, -ci, cc, br, bg, bo, bz, ck, -ca,-bp, -bf, -bq, -cb, cj, by, bn, bh, bs, cd, -ch, -bw, -bl, -bj, -bu,-cf, cf, bu, bj, bl, bw, ch, -cd, -bs, -bh, -bn, -by, -cj, cb, bq, bf,bp, ca, -ck, -bz, -bo, -bg, -br, -cc, ci, bx, bm, bi, bt, ce, -cg, -bv,-bk}{cw, dt, eq, -eg, -dj, -cm, -dg, -ed, et, dw, cz, ct, dq, en, -ej, -dm,-cp, -dd, -ea, ew, dz, dc, cq, dn, ek, -em, -dp, -cs, -da, -dx, -eu, ec,df, cn, dk, eh, -ep, -ds, -cv, -cx, -du, -er, ef, di, cl, dh, ee, -es,-dv, -cy, -cu, -dr, -eo, ei, dl, co, de, eb, -ev, -dy, -db, -cr, -d_,-el}{ai, al, ao, -am, -aj, -ah, -ak, -an, an, ak, ah, aj, am, -ao, -al, -ai,-ai, -al, -ao, am, aj, ah, ak, an, -an, -ak, -ah, -aj, -am, ao, al, ai,ai, al, ao, -am, -aj, -ah, -ak, -an, an, ak, ah, aj, am, -ao, -al, -ai,-ai, -al, -ao, am, aj, ah, ak, an, -an, -ak, -ah, -aj, -am, ao, al, ai}{cx, dw, ev, -dz, -da, -cu, -dt, -es, ec, dd, cr, dq, ep, -ef, -dg, -co,-dn, -em, ei, dj, cl, dk, ej, -el, -dm, -cn, -dh, -eg, eo, dp, cq, de,ed, -er, -ds, -ct, -db, -ea, eu, dv, cw, cy, dx, ew, -dy, -cz, -cv, -du,-et, eb, dc, cs, dr, eq, -ee, -df, -cp, -d_, -en, eh, di, cm, dl, ek}{bl, by, -ck, -bx, -bk, -bm, -bz, cj, bw, bj, bn, ca, -ci, -bv, -bi,-bo, -cb, ch, bu, bh, bp, cc, -cg, -bt, -bg, -bq, -cd, cf, bs, bf, br,ce, -ce, -br, -bf, -bs, -cf, cd, bq, bg, bt, cg, -cc, -bp, -bh, -bu,-ch, cb, bo, bi, bv, ci, -ca, -bn, -bj, -bw, -cj, bz, bm, bk, bx, ck,-by, -bl}{cy, dz, -et, -ds, -cr, -df, -eg, em, dl, cl, dm, en, -ef, -de, -cs,-dt, -eu, dy, cx, cz, ea, -es, -dr, -cq, -dg, -eh, el, dk, cm, dn, eo,-ee, -dd, -ct, -du, -ev, dx, cw, da, eb, -er, -dq, -cp, -dh, -ei, ek,dj, cn, d_, ep, -ed, -dc, -cu, -dv, -ew, dw, cv, db, ec, -eq, -dp, -co,-di, -ej}{as, az, -bd, -aw, -ap, -av, -bc, ba, at, ar, ay, -be, -ax, -aq, -au,-bb, bb, au, aq, ax, be, -ay, -ar, -at, -ba, bc, av, ap, aw, bd, -az,-as, -as, -az, bd, aw, ap, av, bc, -ba, -at, -ar, -ay, be, ax, aq, au,bb, -bb, -au, -aq, -ax, -be, ay, ar, at, ba, -bc, -av, -ap, -aw, -bd,az, as}{cz, ec, -eo, -dl, -cn, -dq, -et, dx, cu, de, eh, -ej, -dg, -cs, -dv,ev, ds, cp, dj, em, -ee, -db, -cx, -ea, eq, dn, cl, d_, er, -dz, -cw,-dc, -ef, el, di, cq, dt, ew, -du, -cr, -dh, -ek, eg, dd, cv, dy, -es,-dp, -cm, -dm, -ep, eb, cy, da, ed, -en, -dk, -co, -dr, -eu, dw, ct, df,ei}{bm, cb, -cf, -bq, -bi, -bx, cj, bu, bf, bt, ci, -by, -bj, -bp, -ce, cc,bn, bl, ca, -cg, -br, -bh, -bw, ck, bv, bg, bs, ch, -bz, -bk, -bo, -cd,cd, bo, bk, bz, -ch, -bs, -bg, -bv, -ck, bw, bh, br, cg, -ca, -bl, -bn,-cc, ce, bp, bj, by, -ci, -bt, -bf, -bu, -cj, bx, bi, bq, cf, -cb, -bm}{da, ef, -ej, -de, -cw, -eb, en, di, cs, dx, -er, -dm, -co, -dt, ev, dq,cl, dp, eu, -du, -cp, -dl, -eq, dy, ct, dh, em, -ec, -cx, -dd, -ei, eg,db, cz, ee, -ek, -df, -cv, -ea, eo, dj, cr, dw, -es, -dn, -cn, -ds, ew,dr, cm, d_, et, -dv, -cq, -dk, -ep, dz, cu, dg, el, -ed, -cy, -dc, -eh}{ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab,ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab,ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab,ab, ac, -ac, -ab, -ab, -ac, ac, ab, ab, ac, -ac, -ab, -ab, -ac, ac, ab}{db, ei, -ee, -cx, -df, -em, ea, ct, dj, eq, -dw, -cp, -dn, -eu, ds, cl,dr, -ev, -d_, -co, -dv, er, dk, cs, dz, -en, -dg, -cw, -ed, ej, dc, da,eh, -ef, -cy, -de, -el, eb, cu, di, ep, -dx, -cq, -dm, -et, dt, cm, dq,-ew, -dp, -cn, -du, es, dl, cr, dy, -eo, -dh, -cv, -ec, ek, dd, cz, eg}{bn, ce, -ca, -bj, -br, -ci, bw, bf, bv, -cj, -bs, -bi, -bz, cf, bo, bm,cd, -cb, -bk, -bq, -ch, bx, bg, bu, -ck, -bt, -bh, -by, cg, bp, bl, cc,-cc, -bl, -bp, -cg, by, bh, bt, ck, -bu, -bg, -bx, ch, bq, bk, cb, -cd,-bm, -bo, -cf, bz, bi, bs, cj, -bv, -bf, -bw, ci, br, bj, ca, -ce, -bn}{dc, el, -dz, -cq, -d_, ew, dn, cr, ea, -ek, -db, -dd, -em, dy, cp, dp,-ev, -dm, -cs, -eb, ej, da, de, en, -dx, -co, -dq, eu, dl, ct, ec, -ei,-cz, -df, -eo, dw, cn, dr, -et, -dk, -cu, -ed, eh, cy, dg, ep, -dv, -cm,-ds, es, dj, cv, ee, -eg, -cx, -dh, -eq, du, cl, dt, -er, -di, -cw, -ef}{at, bc, -ay, -ap, -ax, bd, au, as, bb, -az, -aq, -aw, be, av, ar, ba,-ba, -ar, -av, -be, aw, aq, az, -bb, -as, -au, -bd, ax, ap, ay, -bc,-at, -at, -bc, ay, ap, ax, -bd, -au, -as, -bb, az, aq, aw, -be, -av,-ar, -ba, ba, ar, av, be, -aw, -aq, -az, bb, as, au, bd, -ax, -ap, -ay,bc, at}{dd, eo, -du, -cm, -dx, el, da, dg, er, -dr, -cp, -ea, ei, cx, dj, eu,-d_, -cs, -ed, ef, cu, dm, -ew, -dl, -cv, -eg, ec, cr, dp, -et, -di,-cy, -ej, dz, co, ds, -eq, -df, -db, -em, dw, cl, dv, -en, -dc, -de,-ep, dt, cn, dy, -ek, -cz, -dh, -es, dq, cq, eb, -eh, -cw, -dk, -ev, dn,ct, ee}{bo, ch, -bv, -bh, -ca, cc, bj, bt, -cj, -bq, -bm, -cf, bx, bf, by, -ce,-bl, -br, -ck, bs, bk, cd, -bz, -bg, -bw, cg, bn, bp, ci, -bu, -bi, -cb,cb, bi, bu, -ci, -bp, -bn, -cg, bw, bg, bz, -cd, -bk, -bs, ck, br, bl,ce, -by, -bf, -bx, cf, bm, bq, cj, -bt, -bj, -cc, ca, bh, bv, -ch, -bo}{de, er, -dp, -ct, -eg, ea, cn, dv, -el, -cy, -dk, ew, dj, cz, em, -du,-co, -eb, ef, cs, dq, -eq, -dd, -df, -es, d_, cu, eh, -dz, -cm, -dw, ek,cx, dl, -ev, -di, -da, -en, dt, cp, ec, -ee, -cr, -dr, ep, dc, dg, et,-dn, -cv, -ei, dy, cl, dx, -ej, -cw, -dm, eu, dh, db, eo, -ds, -cq, -ed}{aj, ao, -ak, -ai, -an, al, ah, am, -am, -ah, -al, an, ai, ak, -ao, -aj,-aj, -ao, ak, ai, an, -al, -ah, -am, am, ah, al, -an, -ai, -ak, ao, aj,aj, ao, -ak, -ai, 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cm, -cx, di, -dt, ee, -ep, -et, ei, -dx, dm, -db, cq}{ci, -cd, by, -bt, bo, -bj, bf, -bk, bp, -bu, bz, -ce, cj, ch, -cc, bx,-bs, bn, -bi, bg, -bl, bq, -bv, ca, -cf, ck, cg, -cb, bw, -br, bm, -bh,bh, -bm, br, -bw, cb, -cg, -ck, cf, -ca, bv, -bq, bl, -bg, bi, -bn, bs,-bx, cc, -ch, -cj, ce, -bz, bu, -bp, bk, -bf, bj, -bo, bt, -by, cd, -ci}{es, -ej, ea, -dr, di, -cz, cq, -co, cx, -dg, dp, -dy, eh, -eq, -eu, el,-ec, dt, -dk, db, -cs, cm, -cv, de, -dn, dw, -ef, eo, ew, -en, ee, -dv,dm, -dd, cu, -cl, ct, -dc, dl, -du, ed, -em, ev, ep, -eg, dx, -d_, df,-cw, cn, -cr, da, -dj, ds, -eb, ek, -et, -er, ei, -dz, dq, -dh, cy, -cp}{ao, -an, am, -al, ak, -aj, ai, -ah, ah, -ai, aj, -ak, al, -am, an, -ao,-ao, an, -am, al, -ak, aj, -ai, ah, -ah, ai, -aj, ak, -al, am, -an, ao,ao, -an, am, -al, ak, -aj, ai, -ah, ah, -ai, aj, -ak, al, -am, an, -ao,-ao, an, -am, al, -ak, aj, -ai, ah, -ah, ai, -aj, ak, -al, am, -an, ao}{et, -em, ef, -dy, dr, -dk, dd, -cw, cp, -cn, cu, -db, di, -dp, dw, -ed,ek, -er, -ev, eo, -eh, ea, -dt, dm, -df, cy, -cr, cl, -cs, cz, -dg, dn,-du, eb, -ei, ep, -ew, -eq, ej, -ec, dv, -d_, dh, -da, ct, -cm, cq, -cx,de, -dl, ds, -dz, eg, -en, eu, es, -el, ee, -dx, dq, -dj, dc, -cv, co}{cj, -cg, cd, -ca, bx, -bu, br, -bo, bl, -bi, bf, -bh, bk, -bn, bq, -bt,bw, -bz, cc, -cf, ci, ck, -ch, ce, -cb, by, -bv, bs, -bp, bm, -bj, bg,-bg, bj, -bm, bp, -bs, bv, -by, cb, -ce, ch, -ck, -ci, cf, -cc, bz, -bw,bt, -bq, bn, -bk, bh, -bf, bi, -bl, bo, -br, bu, -bx, ca, -cd, cg, -cj}{eu, -ep, ek, -ef, ea, -dv, dq, -dl, dg, -db, cw, -cr, cm, -co, ct, -cy,dd, -di, dn, -ds, dx, -ec, eh, -em, er, -ew, -es, en, -ei, ed, -dy, dt,-d_, dj, -de, cz, -cu, cp, -cl, cq, -cv, da, -df, dk, -dp, du, -dz, ee,-ej, eo, -et, -ev, eq, -el, eg, -eb, dw, -dr, dm, -dh, dc, -cx, cs, -cn}{be, -bd, bc, -bb, ba, -az, ay, -ax, aw, -av, au, -at, as, -ar, aq, -ap,ap, -aq, ar, -as, at, -au, av, -aw, ax, -ay, az, -ba, bb, -bc, bd, -be,-be, bd, -bc, bb, -ba, az, -ay, ax, -aw, av, -au, at, -as, ar, -aq, ap,-ap, aq, -ar, as, -at, au, -av, aw, -ax, ay, -az, ba, -bb, bc, -bd, be}{ev, -es, ep, -em, ej, -eg, ed, -ea, dx, -du, dr, -d_, dl, -di, df, -dc,cz, -cw, ct, -cq, cn, -cl, co, -cr, cu, -cx, da, -dd, dg, -dj, dm, -dp,ds, -dv, dy, -eb, ee, -eh, ek, -en, eq, -et, ew, eu, -er, eo, -el, ei,-ef, ec, -dz, dw, -dt, dq, -dn, dk, -dh, de, -db, cy, -cv, cs, -cp, cm}{ck, -cj, ci, -ch, cg, -cf, ce, -cd, cc, -cb, ca, -bz, by, -bx, bw, -bv,bu, -bt, bs, -br, bq, -bp, bo, -bn, bm, -bl, bk, -bj, bi, -bh, bg, -bf,bf, -bg, bh, -bi, bj, -bk, bl, -bm, bn, -bo, bp, -bq, br, -bs, bt, -bu,bv, -bw, bx, -by, bz, -ca, cb, -cc, cd, -ce, cf, -cg, ch, -ci, cj, -ck}{ew, -ev, eu, -et, es, -er, eq, -ep, eo, -en, em, -el, ek, -ej, ei, -eh,eg, -ef, ee, -ed, ec, -eb, ea, -dz, dy, -dx, dw, -dv, du, -dt, ds, -dr,dq, -dp, d_, -dn, dm, -dl, dk, -dj, di, -dh, dg, -df, de, -dd, dc, -db,da, -cz, cy, -cx, cw, -cv, cu, -ct, cs, -cr, cq, -cp, co, -cn, cm, -cl}.

In addition to DCT-2 and 4×4 DST-7, which have been employed in HEVC, anAdaptive Multiple Transform (AMT, or as known as Enhanced MultipleTransform (EMT)) scheme has been proposed for residual coding for bothinter and intra coded blocks. The AMT may use multiple selectedtransforms from the DCT/DST families other than the current transformsin HEVC. The newly introduced transform matrices may include DST-7,DCT-8, DST-1 and DCT-5. Table 1 below shows exemplary basis functions ofa selected DST/DCT.

TABLE 1 Transform basis functions of DCT-II/V/VIII and DST-I/VII forN-point input Transform Type Basis function T_(i)(j), i, j = 0, 1, . . ., N − 1 DCT-2 $\quad\begin{matrix}{{T_{i}(j)} = {\omega_{0} \cdot \sqrt{\frac{2}{N}} \cdot {\cos\left( \frac{\pi \cdot i \cdot \left( {{2j} + 1} \right)}{2N} \right)}}} \\{{{where}\mspace{11mu}\omega_{0}} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix} \right.}\end{matrix}$ DCT-5 $\quad\begin{matrix}{{{T_{i}(j)} = {\omega_{0} \cdot \omega_{1} \cdot \sqrt{\frac{2}{{2N} - 1}} \cdot {\cos\left( \frac{2{\pi \cdot i \cdot j}}{{2N} - 1} \right)}}},} \\{{{where}\mspace{11mu}\omega_{0}} = \left\{ {\begin{matrix}\sqrt{\frac{2}{N}} & {i = 0} \\1 & {i \neq 0}\end{matrix},{\omega_{1} = \left\{ \begin{matrix}\sqrt{\frac{2}{N}} & {j = 0} \\1 & {j \neq 0}\end{matrix} \right.}} \right.}\end{matrix}$ DCT-8${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\cos\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {{2j} + 1} \right)}{{4N} + 2} \right)}}$DST-1${T_{i}(j)} = {\sqrt{\frac{2}{N + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {i + 1} \right) \cdot \left( {j + 1} \right)}{N + 1} \right)}}$DST-7${T_{i}(j)} = {\sqrt{\frac{4}{{2N} + 1}} \cdot {\sin\left( \frac{\pi \cdot \left( {{2i} + 1} \right) \cdot \left( {j + 1} \right)}{{2N} + 1} \right)}}$

In order to keep the orthogonality of the transform matrix, thetransform matrices may be quantized more accurately than the transformmatrices in HEVC, with 10-bit representation instead of 8-bit in HEVC.To keep the intermediate values of the transformed coefficients withinthe range of 16-bit, after horizontal and after vertical transform, allthe coefficients may be right shifted by 2 more bits compared to theright shift used in the current HEVC transforms. The AMT may apply tocoding units (CUs) with both width and height smaller than or equal to64, and whether AMT applies or not may be controlled by a CU level flag.When the CU level flag is equal to 0, DCT-2 may be applied in the CU toencode the residue. For luma coding block within an AMT-enabled CU, twoadditional flags may be signalled to identify the horizontal andvertical transform to be used.

For intra residue coding, due to the different residual statistics ofdifferent intra prediction modes, a mode-dependent transform candidateselection process may be used. Three transform subsets may be defined asshown in

Table 2 below, and the transform subset may be selected based on theintra prediction mode, as specified in

Table 3 below.

TABLE 2 Three pre-defined transform candidate sets Transform TransformSet Candidates 0 DST-7 DCT-8 1 DST-7, DST-1 2 DST-7, DCT-5

With the subset concept, a transform subset may be first identifiedbased on Table 3 using the intra prediction mode of a CU with a CU-levelAMT flag that is equal to 1. After that, for each of the horizontaltransform and the vertical transform, one of the two transformcandidates in the identified transform subset, according to Table 2, maybe selected based on being explicitly signalled with flags.

TABLE 3 Selected (H)orizontal and (V)ertical transform sets for eachintra prediction mode Intra Mode 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1516 17 V 2 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 H 2 1 0 1 0 1 0 1 0 1 0 1 01 2 2 2 2 Intra Mode 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34V 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 H 2 2 2 2 2 1 0 1 0 1 0 1 0 1 0 1 0Intra Mode 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 V 1 0 10 1 0 1 0 1 0 1 2 2 2 2 2 2 2 H 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0Intra Mode 53 54 55 56 57 58 59 60 61 62 63 64 65 66 V 2 2 1 0 1 0 1 0 10 1 0 1 0 H 0 0 1 0 1 0 1 0 1 0 1 0 1 0

However, for inter prediction residual, only one transform set, whichconsists of DST-7 and DCT-8, may be used for all inter modes and forboth horizontal and vertical transforms. Among the four additionaltransform types, i.e., DST-7, DCT-8, DST-1 and DCT-5, the mostefficiently used transform types are DST-7 and DCT-8. It is noted thatDCT-8 is essentially flipping DST-7 basis left and right with signchanges, so DCT-8 and DST-7 may basically share the same transformbases.

The transform core, which is a matrix composed by the basis vectors, ofDST-7 can be also be represented as shown below:

4−point DST−7: a, b, c, d c, c, 0, −c d, −a, −c, b b, −d, c, −a 8−pointDST−7: a, b, c, d, e, f, g, h c, f, h, e, b, −a, −d, −g e, g, b, −c, −h,−d, a, f g, c, −d, −f a, h, b, −e h, −a, −g, b, f, −c, −e, d f, −e, −a,g, −d, −b, h, −c d, −h, e, −a, −c, g, −f, b b, −d, f, −h, g, −e, c, −a16−point DST−7: a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p c, f, i,l, o, o, l, i, f, c, 0, −c, −f, −i, −l, −o e, j, o, m, h, c, −b, −g, −l,−p, −k, −f −a, d, i, n g, n, l, e, −b, −i, −p, −j, −c, d, k, o, h, a,−f, −m i, o, f, −c, −l, −l, −c, f, o, i, 0, −i, −o, −f, c, l k, k, 0,−k, −k, 0, k, k, 0, −k, −k, 0, k, k, 0, −k m, g, −f, −n, −a, l, h, −e,−o, −b, k, i, −d, −p, −c, j o, c, −l, −f, i, i, −f, −l, c, o, 0, −o, −c,l, f, −i p, −a, −o, b, n, −c, −m, d, l, −e, −k, f, j, −g, −i, h n, −e,−i, j, d, −o, a, m, −f, −h, k, c, −p, b, l, −g i, −i, −c, o, −f, −f, o,−c, −i, l, 0, −l, i, c, −o, f j, −m, c, g, −p, f, d, −n, i, a, −k, l,−b, −h, o, −e h, −p, i, −a, −g, o, −j, b, f, −n, k, −c, −e, m, −l, d f,−l, o, −i, c, c, −i, o, −l, f, 0, −f, l, −o, i, −c d, −h, l, −p, m, −i,e, −a, −c, g, −k, o, −n, j, −f, b b, −d, f, −h, j, −l, n, −p, o, −m, k,−i, g, −e, c, −a 32−point DST−7: a, b, c, d, e, f, g, h, i, j, k, l, m,n, o, p, q, r, s, t, u, v, w, x, y, z, A, B, C, D, E, F c, f, i, l, o,r, u, x, A, D, F, C, z, w, t, q, n, k, h, e, b, −a, −d, −g, −j, −m, −p,−s, −v, −y, −B, −E e, j, o, t, y, D, D, y, t, o, j, e, 0, −e, −j, −o,−t, −y, −D, −D, −y, −t, −o, −j, −e, 0, e, j, o, t, y, D g, n, u, B, D,w, p, i, b, −e, −l, −s, −z, −F, −y, −r, −k, −d, c, j, q, x, E, A, t, m,f, −a, −h, −o, −v, −C i, r, A, C, t, k, b, −g, −p, −y, −E, −v, −m, −d,e, n, w, F, x, o, f, −c, −l, −u, −D, −z, −q, −h, a, j, s, B k, v, F, u,j, −a, −l, −w, −E, −t, −i, b, m, x, D, s, h, −c, −n, −y, −C, −r, −g, d,o, z, B, q, f, −e, −p, −A m, z, z, m, 0, −m, −z, −z, −m, 0, m, z, z, m,0, −m, −z, −z, −m, 0, m, z, z, m, 0, −m, −z, −z, −m, 0, m, z o, D, t, e,−j, −y, −y, −j, e, t, D, o, 0, −o, −D, −t, −e, j, y, y, j, −e, −t, −D,−o, 0, o, D, t, e, −j, −y q, E, n, −c, −t, −B, −k, f, w, y, h, −i, −z,−v, −e, l, C, s, b, −o, −F, −p, a, r, D, m, −d, −u, −A, −j, g, x s, A,h, −k, −D, −p, c, v, x, e, −n, −F, −m, f, y, u, b, −q, −C, −j, i, B, r,−a, −t, −z, −g, l, E, o, −d, −w u, w, b, −s, −y, −d, q, A, f, −o, −C,−h, m, E, j, −k, −F, −l, i, D, n, −g, −B, −p, e, z, r, −c, −x, −t, a, vw, s, −d, −A, −o, h, E, k, −l, −D, −g, p, z, c, −t, −v, a, x, r, −e, −B,−n, i, F, j, −m, −C, −f, q, y, b, −u y, o, −j, −D, −e, t, t, −e, −D, −j,o, y, 0, −y, −o, j, D, e, −t, −t, e, D, j, −o, −y, 0, y, o, −j, −D, −e,t A, k, −p, −v, e, F, f, −u, −q, j, B, a, −z, −l, o, w, −d, −E, −g, t,r, −i, −C, −b, y, m, −n, −x, c, D, h, −s C, g, −v, −n, o, u, −h, −B, a,D, f, −w, −m, p, t, −i, −A, b, E, e, −x, −l, q, s, −j, −z, c, F, d, −y,−k, r E, c, −B, −f, y, i, −v, −l, s, o, −p, −r, m, u, −j, −x, g, A, −d,−D, a, F, b, −C, −e, z, h, −w, −k, t, n, −q F, −a, −E, b, D, −c, −C, d,B, −e, −A, f, z, −g, −y, h, x, −i, −w, j, v, −k, −u, l, t, −m, −s, n, r,−o, −q, p D, −e, −y, j, t, −o, −o, t, j, −y, −e, D, 0, −D, e, y, −j, −t,o, o, −t, −j, y, e, −D, 0, D, −e, −y, j, t, −o B, −i, −s, r, j, −A, −a,C, −h, −t, q, k, −z, −b, D, −g, −u, p, l, −y, −c, E, −f, −v, o, m, −x,−d, F, −e, −w, n z, −m, −m, z, 0, −z, m, m, −z, 0, z, −m, −m, z, 0, −z,m, m, −z, 0, z, −m, −m, z, 0, −z, m, m, −z, 0, z, −m x, −q, −g, E, −j,−n, A, −c, −u, t, d, −B, m, k, −D, f, r, −w, −a, y, −p, −h, F, −i, −o,z, −b, −v, s, e, −C, l v, −u, −a, w, −t, −b, x, −s, −c, y, −r, −d, z,−q, −e, A, −p, −f, B, −o, −g, C, −n, −h, D, −m, −i, E, −l, −j, F, −k t,−y, e, o, −D, j, j, −D, o, e, −y, t, 0, −t, y, −e, −o, D, −j, −j, D, −o,−e, y, −t, 0, t, −y, e, o, −D, j r, −C, k, g, −y, v, −d, −n, F, −o, −c,u, −z, h, j, −B, s, −a, −q, D, −l, −f, x, −w, e, m, −E, p, b, −t, A, −ip, −F, q, −a, −o, E, −r, b, n, −D, s, −c, −m, C, −t, d, l, −B, u, −e,−k, A, −v, f, j, −z, w, −g, −i, y, −x, h n, −B, w, −i, −e, s, −F, r, −d,−j, x, −A, m, a, −o, C, −v, h, f, −t, E, −q, c, k, −y, z, −l, −b, p, −D,u, −g l, −x, C, −q, e, g, −s, E, −v, j, b, −n, z, −A, o, −c, −i, u, −F,t, −h, −d, p, −B, y, −m, a, k, −w, D, −r, f j, −t, D, −y, o, −e, −e, o,−y, D, −t, j, 0, −j, t, −D, y, −o, e, e, −o, y, −D, t, −j, 0, j, −t, D,−y, o, −e h, −p, x, −F, y, −q, i, −a, −g, o, −w, E, −z, r, −j, b, f, −n,v, −D, A, −s, k, −c, −e, m, −u, C, −B, t, −l, d f, −l, r, −x, D, −C, w,−q, k, −e, −a, g, −m, s, −y, E, −B, v, −p, j, −d, −b, h, −n, t, −z, F,−A, u, −o, i, −c d, −h, l, −p, t, −x, B, −F, C, −y, u, −q, m, −i, e, −a,−c, g, −k, o, −s, w, −A, E, −D, z, −v, r, −n, j, −f, b b, −d, f, −h, j,−l, n, −p, r, −t, v, −x, z, −B, D, −F, E, −C, A, −y, w, −u, s, −q, o,−m, k, −i, g, −e, c, −a

64-point DST-7:

{aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar,as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj,bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb,cc, cd, ce, cf, cg, ch, ci, cj, ck, cl,}{ab, ae, ah, ak, an, aq, at, aw, az, bc, bf, bi, bl, bo, br, bu, bx, ca,cd, cg, cj, cl, ci, cf, cc, bz, bw, bt, bq, bn, bk, bh, be, bb, ay, av,as, ap, am, aj, ag, ad, aa, -ac, -af, -ai, -al, -ao, -ar, -au, -ax, -ba,-bd, -bg, -bj, -bm, -bp, -bs, -bv, -by, -cb, -ce, -ch, -ck,}{ac, ah, am, ar, aw, bb, bg, bl, bq, bv, ca, cf, ck, ci, cd, by, bt, bo,bj, be, az, au, ap, ak, af, aa, -ae, -aj, -ao, -at, -ay, -bd, -bi, -bn,-bs, -bx, -cc, -ch, -cl, -cg, -cb, -bw, -br, -bm, -bh, -bc, -ax, -as,-an, -ai, -ad, ab, ag, al, aq, av, ba, bf, bk, bp, bu, bz, ce, cj,}{ad, ak, ar, ay, bf, bm, bt, ca, ch, cj, cc, bv, bo, bh, ba, at, am, af,-ab, -ai, -ap, -aw, -bd, -bk, -br, -by, -cf, -cl, -ce, -bx, -bq, -bj,-bc, -av, -ao, -ah, -aa, ag, an, au, bb, bi, bp, bw, cd, ck, cg, bz, bs,bl, be, ax, aq, aj, ac, -ae, -al, -as, -az, -bg, -bn, -bu, -cb, -ci,}{ae, an, aw, bf, bo, bx, cg, ci, bz, bq, bh, ay, ap, ag, -ac, -al, -au,-bd, -bm, -bv, -ce, -ck, -cb, -bs, -bj, -ba, -ar, -ai, aa, aj, as, bb,bk, bt, cc, cl, cd, bu, bl, bc, at, ak, ab, -ah, -aq, -az, -bi, -br,-ca, -cj, -cf, -bw, -bn, -be, -av, -am, -ad, af, ao, ax, bg, bp, by,ch,}{af, aq, bb, bm, bx, ci, ce, bt, bi, ax, am, ab, -aj, -au, -bf, -bq,-cb, -cl, -ca, -bp, -be, -at, -ai, ac, an, ay, bj, bu, cf, ch, bw, bl,ba, ap, ae, -ag, -ar, -bc, -bn, -by, -cj, -cd, -bs, -bh, -aw, -al, -aa,ak, av, bg, br, cc, ck, bz, bo, bd, as, ah, -ad, -ao, -az, -bk, -bv,-cg,}{ag, at, bg, bt, cg, ce, br, be, ar, ae, -ai, -av, -bi, -bv, -ci, -cc,-bp, -bc, -ap, -ac, ak, ax, bk, bx, ck, ca, bn, ba, an, aa, -am, -az,-bm, -bz, -cl, -by, -bl, -ay, -al, ab, ao, bb, bo, cb, cj, bw, bj, aw,aj, -ad, -aq, -bd, -bq, -cd, -ch, -bu, -bh, -au, -ah, af, as, bf, bs,cf,}{ah, aw, bl, ca, ci, bt, be, ap, aa, -ao, -bd, -bs, -ch, -cb, -bm, -ax,-ai, ag, av, bk, bz, cj, bu, bf, aq, ab, -an, -bc, -br, -cg, -cc, -bn,-ay, -aj, af, au, bj, by, ck, bv, bg, ar, ac, -am, -bb, -bq, -cf, -cd,-bo, -az, -ak, ae, at, bi, bx, cl, bw, bh, as, ad, -al, -ba, -bp, -ce,}{ai, az, bq, ch, bz, bi, ar, aa, -aq, -bh, -by, -ci, -br, -ba, -aj, ah,ay, bp, cg, ca, bj, as, ab, -ap, -bg, -bx, -cj, -bs, -bb, -ak, ag, ax,bo, cf, cb, bk, at, ac, -ao, -bf, -bw, -ck, -bt, -bc, -al, af, aw, bn,ce, cc, bl, au, ad, -an, -be, -bv, -cl, -bu, -bd, -am, ae, av, bm, cd,}{aj, bc, bv, cj, bq, ax, ae, -ao, -bh, -ca, -ce, -bl, -as, aa, at, bm,cf, bz, bg, an, -af, -ay, -br, -ck, -bu, -bb, -ai, ak, bd, bw, ci, bp,aw, ad, -ap, -bi, -cb, -cd, -bk, -ar, ab, au, bn, cg, by, bf, am, -ag,-az, -bs, -cl, -bt, -ba, -ah, al, be, bx, ch, bo, av, ac, -aq, -bj,-cc,}{ak, bf, ca, cc, bh, am, -ai, -bd, -by, -ce, -bj, -ao, ag, bb, bw, cg,bl, aq, -ae, -az, -bu, -ci, -bn, -as, ac, ax, bs, ck, bp, au, -aa, -av,-bq, -cl, -br, -aw, -ab, at, bo, cj, bt, ay, ad, -ar, -bm, -ch, -bv,-ba, -af, ap, bk, cf, bx, bc, ah, -an, -bi, -cd, -bz, -be, -aj, al, bg,cb,}{al, bi, cf, bv, ay, ab, -av, -bs, -ci, -bl, -ao, ai, bf, cc, by, bb,ae, -as, -bp, -cl, -bo, -ar, af, bc, bz, cb, be, ah, -ap, -bm, -cj, -br,-au, ac, az, bw, ce, bh, ak, -am, -bj, -cg, -bu, -ax, -aa, aw, bt, ch,bk, an, -aj, -bg, -cd, -bx, -ba, -ad, at, bq, ck, bn, aq, -ag, -bd,-ca,}{am, bl, ck, bo, ap, -aj, -bi, -ch, -br, -as, ag, bf, ce, bu, av, -ad,-bc, -cb, -bx, -ay, aa, az, by, ca, bb, ac, -aw, -bv, -cd, -be, -af, at,bs, cg, bh, ai, -aq, -bp, -cj, -bk, -al, an, bm, cl, bn, ao, -ak, -bj,-ci, -bq, -ar, ah, bg, cf, bt, au, -ae, -bd, -cc, -bw, -ax, ab, ba, bz,}{an, bo, ci, bh, ag, -au, -bv, -cb, -ba, aa, bb, cc, bu, at, -ah, -bi,-cj, -bn, -am, ao, bp, ch, bg, af, -av, -bw, -ca, -az, ab, bc, cd, bt,as, -ai, -bj, -ck, -bm, -al, ap, bq, cg, bf, ae, -aw, -bx, -bz, -ay, ac,bd, ce, bs, ar, -aj, -bk, -cl, -bl, -ak, aq, br, cf, be, ad, -ax, -by,}{ao, br, cd, ba, -ac, -bf, -ci, -bm, -aj, at, bw, by, av, -ah, -bk, -ck,-bh, 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-ay, ak, bt, bv, am, -aw, -cf, -bj, -aa, bi, cg, ax, -al,-bu,}{as, cd, bj, -ab, -bm, -ca, -ap, av, cg, bg, -ae, -bp, -bx, -am, ay, cj,bd, -ah, -bs, -bu, -aj, bb, cl, ba, -ak, -bv, -br, -ag, be, ci, ax, -an,-by, -bo, -ad, bh, cf, au, -aq, -cb, -bl, -aa, bk, cc, ar, -at, -ce,-bi, ac, bn, bz, ao, -aw, -ch, -bf, af, bq, bw, al, -az, -ck, -bc, ai,bt,}{at, cg, be, -ai, -bv, -bp, -ac, bk, ca, an, -az, -cl, -ay, ao, cb, bj,-ad, -bq, -bu, -ah, bf, cf, as, -au, -ch, -bd, aj, bw, bo, ab, -bl, -bz,-am, ba, ck, ax, -ap, -cc, -bi, ae, br, bt, ag, -bg, -ce, -ar, av, ci,bc, -ak, -bx, -bn, -aa, bm, by, al, -bb, -cj, -aw, aq, cd, bh, -af,-bs,}{au, cj, az, -ap, -ce, -be, ak, bz, bj, -af, -bu, -bo, aa, bp, bt, ae,-bk, -by, -aj, bf, cd, ao, -ba, -ci, -at, av, ck, ay, -aq, -cf, -bd, al,ca, bi, -ag, -bv, -bn, ab, bq, bs, ad, -bl, -bx, -ai, bg, cc, an, -bb,-ch, -as, aw, cl, ax, -ar, -cg, -bc, am, cb, bh, -ah, -bw, -bm, ac, br,}{av, cl, au, -aw, -ck, -at, ax, cj, as, -ay, -ci, -ar, az, ch, aq, -ba,-cg, -ap, bb, cf, ao, -bc, -ce, -an, bd, cd, am, -be, -cc, -al, bf, cb,ak, -bg, -ca, -aj, bh, bz, ai, -bi, -by, -ah, bj, bx, ag, -bk, -bw, -af,bl, bv, ae, -bm, -bu, -ad, bn, bt, ac, -bo, -bs, -ab, bp, br, aa, -bq,}{aw, ci, ap, -bd, -cb, -ai, bk, bu, ab, -br, -bn, af, by, bg, -am, -cf,-az, at, cl, as, -ba, -ce, -al, bh, bx, ae, -bo, -bq, ac, bv, bj, -aj,-cc, -bc, aq, cj, av, -ax, -ch, -ao, be, ca, ah, -bl, -bt, -aa, bs, bm,-ag, -bz, -bf, an, cg, ay, -au, -ck, -ar, bb, cd, ak, -bi, -bw, -ad,bp,}{ax, cf, ak, -bk, -bs, ac, bx, bf, -ap, -ck, -as, bc, ca, af, -bp, -bn,ah, cc, ba, -au, -ci, -an, bh, bv, aa, -bu, -bi, am, ch, av, -az, -cd,-ai, bm, bq, -ae, -bz, -bd, ar, cl, aq, -be, -by, -ad, br, bl, -aj, -ce,-ay, aw, cg, al, -bj, -bt, ab, bw, bg, -ao, -cj, -at, bb, cb, ag, -bo,}{ay, cc, af, -br, -bj, an, ck, aq, -bg, -bu, ac, bz, bb, -av, -cf, -ai,bo, bm, -ak, -ch, -at, bd, bx, aa, -bw, -be, as, ci, al, -bl, -bp, ah,ce, aw, -ba, -ca, -ad, bt, bh, -ap, -cl, -ao, bi, bs, -ae, -cb, -az, ax,cd, ag, -bq, -bk, am, cj, ar, -bf, -bv, ab, by, bc, -au, -cg, -aj, bn,}{az, bz, aa, -by, -ba, ay, ca, ab, -bx, -bb, ax, cb, ac, -bw, -bc, aw,cc, ad, -bv, -bd, av, cd, ae, -bu, -be, au, ce, af, -bt, -bf, at, cf,ag, -bs, -bg, as, cg, ah, -br, -bh, ar, ch, ai, -bq, -bi, aq, ci, aj,-bp, -bj, ap, cj, ak, -bo, -bk, ao, ck, al, -bn, -bl, an, cl, am, -bm,}{ba, bw, -ae, -cf, -ar, bj, bn, -an, -cj, -ai, bs, be, -aw, -ca, aa, cb,av, -bf, -br, aj, ck, am, -bo, -bi, as, ce, ad, -bx, -az, bb, bv, -af,-cg, -aq, bk, bm, -ao, -ci, -ah, bt, bd, -ax, -bz, ab, cc, au, -bg, -bq,ak, cl, al, -bp, -bh, at, cd, ac, -by, -ay, bc, bu, -ag, -ch, -ap, bl,}{bb, bt, -aj, -cl, -ai, bu, ba, -bc, -bs, ak, ck, ah, -bv, -az, bd, br,-al, -cj, -ag, bw, ay, -be, -bq, am, ci, af, -bx, -ax, bf, bp, -an, -ch,-ae, by, aw, -bg, -bo, ao, cg, ad, -bz, -av, bh, bn, -ap, -cf, -ac, ca,au, -bi, -bm, aq, ce, ab, -cb, -at, bj, bl, -ar, -cd, -aa, cc, as, -bk,}{bc, bq, -ao, -ce, aa, cf, an, -br, -bb, bd, bp, -ap, -cd, ab, cg, am,-bs, -ba, be, bo, -aq, 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bb, -bx, -ab, ca, -ay, -ba, by, aa, -bz, az,}{bn, aj, -cg, au, bc, -by, ab, bv, -bf, -ar, cj, -am, -bk, bq, ag, -cd,ax, az, -cb, ae, bs, -bi, -ao, cl, -ap, -bh, bt, ad, -ca, ba, aw, -ce,ah, bp, -bl, -al, ci, -as, -be, bw, aa, -bx, bd, at, -ch, ak, bm, -bo,-ai, cf, -av, -bb, bz, -ac, -bu, bg, aq, -ck, an, bj, -br, -af, cc,-ay,}{bo, ag, -cb, bb, at, -cj, ao, bg, -bw, ab, bt, -bj, -al, cg, -aw, -ay,ce, -aj, -bl, br, ad, -by, be, aq, -cl, ar, bd, -bz, ae, bq, -bm, -ai,cd, -az, -av, ch, -am, -bi, bu, aa, -bv, bh, an, -ci, au, ba, -cc, ah,bn, -bp, -af, ca, -bc, -as, ck, -ap, -bf, bx, -ac, -bs, bk, ak, -cf,ax,}{bp, ad, -bw, bi, ak, -cd, bb, ar, -ck, au, ay, -cg, an, bf, -bz, ag,bm, -bs, -aa, bt, -bl, -ah, ca, -be, -ao, ch, -ax, -av, cj, -aq, -bc,cc, -aj, -bj, bv, -ac, -bq, bo, ae, -bx, bh, al, -ce, ba, as, -cl, at,az, -cf, am, bg, -by, af, bn, -br, -ab, bu, -bk, -ai, cb, -bd, -ap, ci,-aw,}{bq, aa, -br, bp, ab, -bs, bo, ac, -bt, bn, ad, -bu, bm, ae, -bv, bl,af, -bw, bk, ag, -bx, bj, ah, -by, bi, ai, -bz, bh, aj, -ca, bg, ak,-cb, bf, al, -cc, be, am, -cd, bd, an, -ce, bc, ao, -cf, bb, ap, -cg,ba, aq, -ch, az, ar, -ci, ay, as, -cj, ax, at, -ck, aw, au, -cl, av,}{br, -ac, -bm, bw, -ah, -bh, cb, -am, -bc, cg, -ar, -ax, cl, -aw, -as,ch, -bb, -an, cc, -bg, -ai, bx, -bl, -ad, bs, -bq, ab, bn, -bv, ag, bi,-ca, al, bd, -cf, aq, ay, -ck, av, at, -ci, ba, ao, -cd, bf, aj, -by,bk, ae, -bt, bp, -aa, -bo, bu, -af, -bj, bz, -ak, -be, ce, -ap, -az, cj,-au,}{bs, -af, -bh, cd, -aq, -aw, cj, -bb, -al, by, -bm, -aa, bn, -bx, ak,bc, -ci, av, ar, -ce, bg, ag, -bt, br, -ae, -bi, cc, -ap, -ax, ck, -ba,-am, bz, -bl, -ab, bo, -bw, aj, bd, -ch, au, as, -cf, bf, ah, -bu, bq,-ad, -bj, cb, -ao, -ay, cl, -az, -an, ca, -bk, -ac, bp, -bv, ai, be,-cg, at,}{bt, -ai, -bc, ck, -az, -al, bw, -bq, af, bf, -ch, aw, ao, -bz, bn, -ac,-bi, ce, -at, -ar, cc, -bk, -aa, bl, -cb, aq, au, -cf, bh, ad, -bo, by,-an, -ax, ci, -be, -ag, br, -bv, ak, ba, -cl, bb, aj, -bu, bs, -ah, -bd,cj, -ay, -am, bx, -bp, ae, bg, -cg, av, ap, -ca, bm, -ab, -bj, cd, -as,}{bu, -al, -ax, cg, -bi, -aa, bj, -cf, aw, am, -bv, bt, -ak, -ay, ch,-bh, -ab, bk, -ce, av, an, -bw, bs, -aj, -az, ci, -bg, -ac, bl, -cd, au,ao, -bx, br, -ai, -ba, cj, -bf, -ad, bm, -cc, at, ap, -by, bq, -ah, -bb,ck, -be, -ae, bn, -cb, as, aq, -bz, bp, -ag, -bc, cl, -bd, -af, bo, -ca,ar,}{bv, -ao, -as, bz, -br, ak, aw, -cd, bn, -ag, -ba, ch, -bj, ac, be, -cl,bf, ab, -bi, ci, -bb, -af, bm, -ce, ax, aj, -bq, ca, -at, -an, bu, -bw,ap, ar, -by, bs, -al, -av, cc, -bo, ah, az, -cg, bk, -ad, -bd, ck, -bg,-aa, bh, -cj, bc, ae, -bl, cf, -ay, -ai, bp, -cb, au, am, -bt, bx, -aq,}{bw, -ar, -an, bs, -ca, av, aj, -bo, ce, -az, -af, bk, -ci, bd, ab, -bg,cl, -bh, ac, bc, -ch, bl, -ag, -ay, cd, -bp, ak, au, -bz, bt, -ao, -aq,bv, -bx, as, am, -br, cb, -aw, -ai, bn, -cf, ba, ae, -bj, cj, -be, -aa,bf, -ck, bi, -ad, -bb, cg, -bm, ah, ax, -cc, bq, -al, -at, by, -bu, ap,}{bx, -au, -ai, bl, -cj, bg, -ad, -az, cc, -bs, ap, an, -bq, ce, -bb,-ab, be, -ch, bn, -ak, -as, bv, -bz, aw, ag, -bj, cl, -bi, af, ax, -ca,bu, -ar, -al, bo, -cg, bd, -aa, -bc, cf, -bp, am, aq, -bt, cb, -ay, -ae,bh, -ck, bk, -ah, -av, by, -bw, at, aj, -bm, ci, -bf, ac, ba, -cd, br,-ao,}{by, -ax, -ad, be, -cf, br, -aq, -ak, bl, -cl, bk, -aj, -ar, bs, -ce,bd, -ac, -ay, bz, -bx, aw, ae, -bf, cg, -bq, ap, al, -bm, ck, -bj, ai,as, -bt, cd, -bc, ab, az, -ca, bw, -av, -af, bg, -ch, bp, -ao, -am, bn,-cj, bi, -ah, -at, bu, -cc, bb, -aa, -ba, cb, -bv, au, ag, -bh, ci, -bo,an,}{bz, -ba, ab, ax, -bw, cc, -bd, ae, au, -bt, cf, -bg, ah, ar, -bq, ci,-bj, ak, ao, -bn, cl, -bm, an, al, -bk, cj, -bp, aq, ai, -bh, cg, -bs,at, af, -be, cd, -bv, aw, ac, -bb, ca, -by, az, -aa, -ay, bx, -cb, bc,-ad, -av, bu, -ce, bf, -ag, -as, br, -ch, bi, -aj, -ap, bo, -ck, bl,-am,}{ca, -bd, ag, aq, -bn, ck, -bq, at, ad, -ba, bx, -cd, bg, -aj, -an, bk,-ch, bt, -aw, -aa, ax, -bu, cg, -bj, am, ak, -bh, ce, -bw, az, -ac, -au,br, -cj, bm, -ap, -ah, be, -cb, bz, -bc, af, ar, -bo, cl, -bp, as, ae,-bb, by, -cc, bf, -ai, -ao, bl, -ci, bs, -av, -ab, ay, -bv, cf, -bi,al,}{cb, -bg, al, aj, -be, bz, -cd, bi, -an, -ah, bc, -bx, cf, -bk, ap, af,-ba, bv, -ch, bm, -ar, -ad, ay, -bt, cj, -bo, at, ab, -aw, br, -cl, bq,-av, aa, au, -bp, ck, -bs, ax, -ac, -as, bn, -ci, bu, -az, ae, aq, -bl,cg, -bw, bb, -ag, -ao, bj, -ce, by, -bd, ai, am, -bh, cc, -ca, bf, -ak,}{cc, -bj, aq, ac, -av, bo, -ch, bx, -be, al, ah, -ba, bt, -cl, bs, -az,ag, am, -bf, by, -cg, bn, -au, ab, ar, -bk, cd, -cb, bi, -ap, -ad, aw,-bp, ci, -bw, bd, -ak, -ai, bb, -bu, ck, -br, ay, -af, -an, bg, -bz, cf,-bm, at, -aa, -as, bl, -ce, ca, -bh, ao, ae, -ax, bq, -cj, bv, -bc, aj,}{cd, -bm, av, -ae, -am, bd, -bu, cl, -bv, be, -an, -ad, au, -bl, cc,-ce, bn, -aw, af, al, -bc, bt, -ck, bw, -bf, ao, ac, -at, bk, -cb, cf,-bo, ax, -ag, -ak, bb, -bs, cj, -bx, bg, -ap, -ab, as, -bj, ca, -cg, bp,-ay, ah, aj, -ba, br, -ci, by, -bh, aq, aa, -ar, bi, -bz, ch, -bq, az,-ai,}{ce, -bp, ba, -al, -ad, as, -bh, bw, -cl, bx, -bi, at, -ae, -ak, az,-bo, cd, -cf, bq, -bb, am, ac, -ar, bg, -bv, ck, -by, bj, -au, af, aj,-ay, bn, -cc, cg, -br, bc, -an, -ab, aq, -bf, bu, -cj, bz, -bk, av, -ag,-ai, ax, -bm, cb, -ch, bs, -bd, ao, aa, -ap, be, -bt, ci, -ca, bl, -aw,ah,}{cf, -bs, bf, -as, af, ah, -au, bh, -bu, ch, -cd, bq, -bd, aq, -ad, -aj,aw, -bj, bw, -cj, cb, -bo, bb, -ao, ab, al, -ay, bl, -by, cl, -bz, bm,-az, am, aa, -an, ba, -bn, ca, -ck, bx, -bk, ax, -ak, -ac, ap, -bc, bp,-cc, ci, -bv, bi, -av, ai, ae, -ar, be, -br, ce, -cg, bt, -bg, at, -ag,}{cg, -bv, bk, -az, ao, -ad, -ah, as, -bd, bo, -bz, ck, -cc, br, -bg, av,-ak, -aa, al, -aw, bh, -bs, cd, -cj, by, -bn, bc, -ar, ag, ae, -ap, ba,-bl, bw, -ch, cf, -bu, bj, -ay, an, -ac, -ai, at, -be, bp, -ca, cl, -cb,bq, -bf, au, -aj, -ab, am, -ax, bi, -bt, ce, -ci, bx, -bm, bb, -aq, af,}{ch, -by, bp, -bg, ax, -ao, af, ad, -am, av, -be, bn, -bw, cf, -cj, ca,-br, bi, -az, aq, -ah, -ab, ak, -at, bc, -bl, bu, -cd, cl, -cc, bt, -bk,bb, -as, aj, -aa, -ai, ar, -ba, bj, -bs, cb, -ck, ce, -bv, bm, -bd, au,-al, ac, ag, -ap, ay, -bh, bq, -bz, ci, -cg, bx, -bo, bf, -aw, an, -ae,}{ci, -cb, bu, -bn, bg, -az, as, -al, ae, ac, -aj, aq, -ax, be, -bl, bs,-bz, cg, -ck, cd, -bw, bp, -bi, bb, -au, an, -ag, -aa, ah, -ao, av, -bc,bj, -bq, bx, -ce, cl, -cf, by, -br, bk, -bd, aw, -ap, ai, -ab, -af, am,-at, ba, -bh, bo, -bv, cc, -cj, ch, -ca, bt, -bm, bf, -ay, ar, -ak, ad,}{cj, -ce, bz, -bu, bp, -bk, bf, -ba, av, -aq, al, -ag, ab, ad, -ai, an,-as, ax, -bc, bh, -bm, br, -bw, cb, -cg, cl, -ch, cc, -bx, bs, -bn, bi,-bd, ay, -at, ao, -aj, ae, aa, -af, ak, -ap, au, -az, be, -bj, bo, -bt,by, -cd, ci, -ck, cf, -ca, bv, -bq, bl, -bg, bb, -aw, ar, -am, ah, -ac,}{ck, -ch, ce, -cb, by, -bv, bs, -bp, bm, -bj, bg, -bd, ba, -ax, au, -ar,ao, -al, ai, -af, ac, aa, -ad, ag, -aj, am, -ap, as, -av, ay, -bb, be,-bh, bk, -bn, bq, -bt, bw, -bz, cc, -cf, ci, -cl, cj, -cg, cd, -ca, bx,-bu, br, -bo, bl, -bi, bf, -bc, az, -aw, at, -aq, an, -ak, ah, -ae, ab,}{cl, -ck, cj, -ci, ch, -cg, cf, -ce, cd, -cc, cb, -ca, bz, -by, bx, -bw,bv, -bu, bt, -bs, br, -bq, bp, -bo, bn, -bm, bl, -bk, bj, -bi, bh, -bg,bf, -be, bd, -bc, bb, -ba, az, -ay, ax, -aw, av, -au, at, -as, ar, -aq,ap, -ao, an, -am, al, -ak, aj, -ai, ah, -ag, af, -ae, ad, -ac, ab, -aa,}

The variables a, b c, aa, ab, ac, cl can be derived based on theformulations of DST-7 shown in Table 1, and their values can bedifferent for different sizes of DST-7. For example, the value of “a”can be different for 4-point DST-7 and 8-point DST-7. To avoid floatingpoint operations, similar to the DCT-2 cores used in HEVC, the transformcore of DST-7 may be scaled by a pre-defined factor, e.g., 64·√{squareroot over (N)} (corresponds to 8-bit transform core) or 256·√{squareroot over (N)} (corresponds to 10-bit transform core), and rounded tothe nearest integer, or further tuned by an offset, e.g., +1/−1.

In an exemplary design of AMT, when AMT is applied, all primarytransform cores, including DCT-2, may be represented by 10-bit integers,i.e., a 10-bit transform core. However, it may be the case that at leastone of the primary transform cores (but not all) is represented by10-bit integers.

The primary transform in VVC may apply both 8-bit (e.g., used for allother transform sizes except for 64-point and larger) and 10-bit (e.g.,used for 64-point transform and larger) transform cores. In thisapplication, the 2-, 4-, 8-, 16- and 32-point DCT-2 transform corecannot be extracted from, for example, a 64-point DCT-2 transform core,which inevitably increases the cost of applying primary transformbecause a separate 64-point transform core would need extra storage andlogic for implementation. Comparing to 10-bit primary transform cores,the 8-bit primary transform core reduces the complexity because of thereduced memory for storing transform cores and reduced input bit-widthof multiplier/add/sub operations.

Thus, the disclosure provides a set of 8-bit primary transform cores forall transform size types applied in an exemplary AMT including DCT-2,DCT-5, DCT-8, DST-1 and DST-7. The following exemplary proposed methodsmay be used separately or combined in any order.

Referring to FIG. 5, in a first, simple example, a method of utilizingan 8-bit primary transform core matrix to decode compressed video orimage data or encode uncompressed video or image data may include: Step501: receiving information regarding a target file for encoding ordecoding. The target file for encoding or decoding may be one of: acompressed video or image data file or an uncompressed video or imagedata file. The method of FIG. 5 may also include a Step 502 thatincludes determining, for the encoding or decoding of the target file,whether to use a first transform core matrix that is of a first sizetype or a second transform core matrix that is of a second size typethat is smaller than the first size type. The first size type may be the8-bit primary transform core matrix, which has the core matrix that isrepresented by 8-bits and that has a size that is 64-point or larger.The method according to the aspect of the disclosure may further includea Step 503 that includes, when a result of the determination is to usethe first transform core matrix of the first size type, causing ortransmitting information that causes the target file for encoding ordecoding to be encoded or decoded using the 8-bit primary transform corematrix that has the size that is 64-point or larger. The method of FIG.5 may also include a Step 504 that includes, when a result of thedetermination is to use the second transform core matrix of the secondsize type that is smaller than the first size type: extracting thesecond transform core matrix of the second size type from the 8-bitprimary transform core matrix that is 64-point or larger and causing ortransmitting information that causes the target file for encoding ordecoding to be encoded or decoded using the extracted second transformcore matrix of the second type.

The method illustrated in FIG. 5 may also include processing Steps501-504 being performed by one or more processors. The methodillustrated in FIG. 5 may also include storing, by the one or moreprocessors in a memory, the 8-bit primary transform core matrix inadvance before or during the performing of the Steps 501-504.

The method illustrated in FIG. 5 may also include that the firsttransform core matrix is a 64-point, 8-bit primary transform core matrixthat is constructed using thirty-three integers which are used toconstruct a 32-point, 8-bit primary transform core matrix and anadditional thirty-two integers that include: bf, bg, bh, bi, bj, bk, bl,bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb, cc, cd,ce, cf, cg, ch, ci, cj and ck plus their respective sign changes, andthe additional thirty-two integers {bf, bg, bh, bi, bj, bk, bl, bm, bn,bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb, cc, cd, ce, cf,cg, ch, ci, cj, ck} are equal to {91, 90, 90, 90, 88, 87, 86, 84, 83,81, 79, 77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44, 41, 37, 33, 28, 24,20, 15, 11, 7, 2}.

The method illustrated in FIG. 5 may also include that the 128-point8-bit primary transform core is constructed using the 65 integers usedto construct a 64-point 8-bit primary transform core and additional 64integers including: cl, cm, cn, co, cp, cq, cr, cs, ct, cu, cv, cw, cx,cy, cz, da, db, dc, dd, de, df, dg, dh, di, dj, dk, dl, dm, dn, d_, dp,dq, dr, ds, dt, du, dv, dw, dx, dy, dz, ea, eb, ec, ed, ee, ef, eg, eh,ei, ej, ek, el, em, en, eo, ep, eq, er, es, et, eu, ev and ew plus theirsign changes, and these additional 64 integers are {cl, cm, cn, co, cp,cq, cr, cs, ct, cu, cv, cw, cx, cy, cz, da, db, dc, dd, de, df, dg, dh,di, dj, dk, dl, dm, dn, d_, dp, dq, dr, ds, dt, du, dv, dw, dx, dy, dz,ea, eb, ec, ed, ee, ef, eg, eh, ei, ej, ek, el, em, en, eo, ep, eq, er,es, et, eu, ev, ew} are equal to {90, 90, 90, 90, 90, 90, 89, 89, 89,88, 88, 87, 86, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 73, 72,71, 69, 68, 66, 65, 63, 62, 60, 58, 58, 55, 53, 51, 49, 48, 45, 44, 42,40, 38, 36, 34, 32, 29, 27, 25, 23, 21, 19, 17, 15, 12, 10, 8, 6, 3, 1}.

The method illustrated in FIG. 5 may also include that the size of the8-bit primary transform core matrix is 64-point or 128-point. Further,the method illustrated in FIG. 5 may also include that a left half or aright half of even or odd rows of the 8-bit primary transform corematrix form a matrix which is similar to the second transform corematrix, the 8-bit primary transform core matrix is one of: DCT-2, DCT-5,DCT-8, DST-1 and DST-7, and/or the second transform core matrix is oneof: DCT-2, DCT-5, DCT-8, DST-1 and DST-7.

The method illustrated in FIG. 5 may also include that the 8-bit primarytransform core matrix and the second transform core matrix are the sameone of: DCT-2, DCT-5, DCT-8, DST-1 and DST-7. The method illustrated inFIG. 5 may also include that the 8-bit primary transform core matrix isan 8-bit, 64-point DCT-2 transform matrix and/or the second transformcore matrix is an 8-bit, 32-point DCT-2 transform matrix.

The method illustrated in FIG. 5 may also include that the 8-bit,N-point primary transform core matrix, which is composed by M uniquenumbers, is derived by the following steps illustrated in FIG. 6. FIG. 6illustrates a Step 601 that includes deriving an initial transform corematrix by scaling an original transform core using a scaling factor. Forexample, the Step 601 may include deriving an initial transform corematrix T1 by scaling an original transform core which is composed byfloating-point numbers T0 using a scaling factor 2B−2×log 2(N) such thatT1=T0×2^(B-2)×log₂(N), where B=8, and the row vectors of T0, which arebasis vectors, are orthogonal to each other, and the norm of each rowvector is 1.

The method of FIG. 6 may also include a Step 602 that includes derivingan adjusted transform core matrix. For example, Step 602 may includederiving an adjusted transform core matrix (T2) by adding offsets on aselected set of the M unique numbers which are used to construct theadjusted primary transform core matrix T2.

The method of FIG. 6 may also include a Step 603 that includescalculating a cost value C as the sum of absolute values of allelements. For example, the Step 603 may include calculating a cost valueC as the sum of absolute values of all elements of(T1*T1^(T)−2^(2B-4)×log₂(N)²×I), where I is an N×N identity matrix.

The method of FIG. 6 may also include a Step 604 that includes updatingthe adjusted transform matrix. For example, the Step 604 may includeupdating the adjusted transform matrix T2 by adding the calculatedoffset values to T2, which minimizes the cost value to T2, which may beoutput as the 8-bit, N-point primary transform core matrix.

In the method of FIG. 6, the values of the offsets for each of the Munique numbers may include one or more of the following values: −2, −1,0, +1, +2. Moreover, the method of FIG. 6 may also include afterminimizing the cost value C by trying all possible combinations ofoffset values on a selected set of M unique numbers, outputting T2 asthe 8-bit, N-point primary transform core matrix.

Next, the different types of transform cores (transform core matrices)will be explained. As an example, a 64-point DCT-2 transform core may berepresented using 8-bit integers, where the left half of the even rows(row 0, 2, 4, . . . , 30) of the transform core matrix form a matrixwhich is identical to the 32-point DCT-2 transform core matrix. In oneembodiment, the 64-point DCT-2 core matrix may be derived by setting theparameters as follows: {aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al,am, an, ao, ap, aq, ar, as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd,be, bf, bg, bh, bi, bj, bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, bv,bw, bx, by, bz, ca, cb, cc, cd, ce, cf, cg, ch, ci, cj, ck}={64, 83, 36,89, 75, 50, 18, 90, 87, 80, 70, 57, 43, 25, 9, 90, 90, 88, 85, 82, 78,73, 67, 61, 54, 46, 38, 31, 22, 13, 4, 91, 90, 90, 90, 88, 87, 86, 84,83, 81, 79, 77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44, 41, 37, 33, 28,24, 20, 15, 11, 7, 2}.

A 128-point DCT-2 transform core may be represented using 8-bitintegers, and the left half of the even rows (row 0, 2, 4, . . . , 126)of the transform core matrix may form a matrix which is identical to the64-point DCT-2 transform core matrix. In one embodiment, the 128-pointDCT-2 core matrix may be derived by setting the parameters as follows:{aa, ab, ac, ad, ae, af, ag, ah, ai, aj, ak, al, am, an, ao, ap, aq, ar,as, at, au, av, aw, ax, ay, az, ba, bb, bc, bd, be, bf, bg, bh, bi, bj,bk, bl, bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb,cc, cd, ce, cf, cg, ch, ci, cj, ck, cl, cm, cn, co, cp, cq, cr, cs, ct,cu, cv, cw, cx, cy, cz, da, db, dc, dd, de, df, dg, dh, di, dj, dk, dl,dm, dn, d_, dp, dq, dr, ds, dt, du, dv, dw, dx, dy, dz, ea, eb, ec, ed,ee, ef, eg, eh, ei, ej, ek, el, em, en, eo, ep, eq, er, es, et, eu, ev,ew}={64, 83, 36, 89, 75, 50, 18, 90, 87, 80, 70, 57, 43, 25, 9, 90, 90,88, 85, 82, 78, 73, 67, 61, 54, 46, 38, 31, 22, 13, 4, 91, 90, 90, 90,88, 87, 86, 84, 83, 81, 79, 77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44,41, 37, 33, 28, 24, 20, 15, 11, 7, 2, 90, 90, 90, 90, 90, 90, 89, 89,89, 88, 88, 87, 86, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 73,72, 71, 69, 68, 66, 65, 63, 62, 60, 58, 58, 55, 53, 51, 49, 48, 45, 44,42, 40, 38, 36, 34, 32, 29, 27, 25, 23, 21, 19, 17, 15, 12, 10, 8, 6, 3,1}.

A B-bit N-point primary transform core, which is composed by M uniquenumbers (despite the signs and zeros), may be derived by the followingsteps: (a) derive an initial transform core T1 by scaling the originaltransform core which is composed by floating-point numbers, i.e., T0,using a scaling factor 2^(B-2)×log₂(N), i.e., T1=T0×2^(B-2)×log₂(N),where the row vectors of T0, i.e., basis vectors, are orthogonal to eachother, and the norm of each row vector is 1, (b) derive an adjustedtransform core T2 by trying adding offsets on a selected set of M uniquenumbers which are used to construct the primary transform core T2, and(c) calculate the cost value C as the sum of absolute values of allelements of (T1*T1^(T)−2^(2B-4)×log₂(N)²×I), where I is an N×N identitymatrix. The offset values which minimize the cost value may be thenadded to T2, and T2 may be updated. Example offset values for each ofthe M unique numbers may include, but are not limited to: −2, −1, 0, +1,+2. After minimizing the cost value by trying all possible combinationof offset values on a selected set of M unique numbers, T2 may be outputas the B-bit N-point primary transform core.

When deriving the N-point DCT-8 and DST-7 transform cores, it may berestricted that the set of M unique numbers which are used to constructthe N-point DCT-8 core matrix is the same set of M unique numbers whichconstruct the N-point DST-7 core matrix. In one embodiment, M equals toN. In one embodiment, the 8-bit DCT-8 and DST-7 transform cores areshown below. For example, in 4-point DST-7 core, there may be 4 uniquenumbers including 28, 55, 74, 84, and this is the same set of 4 uniquenumbers used to construct 4-point DCT-8 core.

4-Point DST-7:

{29, 55, 74, 84,}

{74, 74, 0, -74,}

{84, -29, -74, 55,}

{55, -84, 74, -29,}

8-Bit DST-7:

{17, 32, 46, 60, 71, 78, 85, 86,}

{46, 78, 86, 71, 32, -17, -60, -85,}

{71, 85, 32, -46, -86, -60, 17, 78,}

{85, 46, -60, -78, 17, 86, 32, -71,}

{86, -17, -85, 32, 78, -46, -71, 60,}

{78, -71, -17, 85, -60, -32, 86, -46,}

{60, -86, 71, -17, -46, 85, -78, 32,}

{32, -60, 78, -86, 85, -71, 46, -17,}

16-Bit DST-7:

{9, 17, 25, 33, 41, 49, 56, 62, 66, 72, 77, 81, 83, 87, 89, 90,}

{25, 49, 66, 81, 89, 89, 81, 66, 49, 25, 0, -25, -49, -66, -81, -89,}

{41, 72, 89, 83, 62, 25, -17, -56, -81, -90, -77, -49, -9, 33, 66, 87,}

{56, 87, 81, 41, -17, -66, -90, -72, -25, 33, 77, 89, 62, 9, -49, -83,}

{66, 89, 49, -25, -81, -81, -25, 49, 89, 66, 0, -66, -89, -49, 25, 81,}

{77, 77, 0, -77, -77, 0, 77, 77, 0, -77, -77, 0, 77, 77, 0, -77,}

{83, 56, -49, -87, -9, 81, 62, -41, -89, -17, 77, 66, -33, -90, -25,72,}

{89, 25, -81, -49, 66, 66, -49, -81, 25, 89, 0, -89, -25, 81, 49, -66,}

{90, -9, -89, 17, 87, -25, -83, 33, 81, -41, -77, 49, 72, -56, -66, 62,}

{87, -41, -66, 72, 33, -89, 9, 83, -49, -62, 77, 25, -90, 17, 81, -56,}

{81, -66, -25, 89, -49, -49, 89, -25, -66, 81, 0, -81, 66, 25, -89, 49,}

{72, -83, 25, 56, -90, 49, 33, -87, 66, 9, -77, 81, -17, -62, 89, -41,}

{62, -90, 66, -9, -56, 89, -72, 17, 49, -87, 77, -25, -41, 83, -81, 33,}

{49, -81, 89, -66, 25, 25, -66, 89, -81, 49, 0, -49, 81, -89, 66, -25,}

{33, -62, 81, -90, 83, -66, 41, -9, -25, 56, -77, 89, -87, 72, -49, 17,}

{17, -33, 49, -62, 72, -81, 87, -90, 89, -83, 77, -66, 56, -41, 25, -9,}

32-Bit DST-7:

{4, 9, 13, 17, 21, 26, 30, 34, 38, 42, 45, 50, 53, 56, 60, 63, 66, 68,72, 74, 77, 78, 80, 82, 84, 85, 86, 88, 88, 89, 90, 90,}

{13, 26, 38, 50, 60, 68, 77, 82, 86, 89, 90, 88, 85, 80, 74, 66, 56, 45,34, 21, 9, -4, -17, -30, -42, -53, -63, -72, -78, -84, -88, -90,}

{21, 42, 60, 74, 84, 89, 89, 84, 74, 60, 42, 21, 0, -21, -42, -60, -74,-84, -89, -89, -84, -74, -60, -42, -21, 0, 21, 42, 60, 74, 84, 89,}

{30, 56, 77, 88, 89, 80, 63, 38, 9, -21, -50, -72, -85, -90, -84, -68,-45, -17, 13, 42, 66, 82, 90, 86, 74, 53, 26, -4, -34, -60, -78, -88,}

{38, 68, 86, 88, 74, 45, 9, -30, -63, -84, -90, -78, -53, -17, 21, 56,80, 90, 82, 60, 26, -13, -50, -77, -89, -85, -66, -34, 4, 42, 72, 88,}

{45, 78, 90, 77, 42, -4, -50, -80, -90, -74, -38, 9, 53, 82, 89, 72, 34,-13, -56, -84, -88, -68, -30, 17, 60, 85, 88, 66, 26, -21, -63, -86,}

{53, 85, 85, 53, 0, -53, -85, -85, -53, 0, 53, 85, 85, 53, 0, -53, -85,-85, -53, 0, 53, 85, 85, 53, 0, -53, -85, -85, -53, 0, 53, 85,}

{60, 89, 74, 21, -42, -84, -84, -42, 21, 74, 89, 60, 0, -60, -89, -74,-21, 42, 84, 84, 42, -21, -74, -89, -60, 0, 60, 89, 74, 21, -42, -84,}

{66, 90, 56, -13, -74, -88, -45, 26, 80, 84, 34, -38, -85, -78, -21, 50,88, 72, 9, -60, -90, -63, 4, 68, 89, 53, -17, -77, -86, -42, 30, 82,}

{72, 86, 34, -45, -89, -63, 13, 78, 82, 21, -56, -90, -53, 26, 84, 77,9, -66, -88, -42, 38, 88, 68, -4, -74, -85, -30, 50, 90, 60, -17, -80,}

{77, 80, 9, -72, -84, -17, 66, 86, 26, -60, -88, -34, 53, 90, 42, -45,-90, -50, 38, 89, 56, -30, -88, -63, 21, 85, 68, -13, -82, -74, 4, 78,}

{80, 72, -17, -86, -60, 34, 90, 45, -50, -89, -30, 63, 85, 13, -74, -78,4, 82, 68, -21, -88, -56, 38, 90, 42, -53, -88, -26, 66, 84, 9, -77,}

{84, 60, -42, -89, -21, 74, 74, -21, -89, -42, 60, 84, 0, -84, -60, 42,89, 21, -74, -74, 21, 89, 42, -60, -84, 0, 84, 60, -42, -89, -21, 74,}

{86, 45, -63, -78, 21, 90, 26, -77, -66, 42, 88, 4, -85, -50, 60, 80,-17, -90, -30, 74, 68, -38, -88, -9, 84, 53, -56, -82, 13, 89, 34, -72,}

{88, 30, -78, -56, 60, 77, -34, -88, 4, 89, 26, -80, -53, 63, 74, -38,-86, 9, 90, 21, -82, -50, 66, 72, -42, -85, 13, 90, 17, -84, -45, 68,}

{90, 13, -88, -26, 84, 38, -78, -50, 72, 60, -63, -68, 53, 77, -42, -82,30, 86, -17, -89, 4, 90, 9, -88, -21, 85, 34, -80, -45, 74, 56, -66,}

{90, -4, -90, 9, 89, -13, -88, 17, 88, -21, -86, 26, 85, -30, -84, 34,82, -38, -80, 42, 78, -45, -77, 50, 74, -53, -72, 56, 68, -60, -66, 63,}

{89, -21, -84, 42, 74, -60, -60, 74, 42, -84, -21, 89, 0, -89, 21, 84,-42, -74, 60, 60, -74, -42, 84, 21, -89, 0, 89, -21, -84, 42, 74, -60,}

{88, -38, -72, 68, 42, -86, -4, 88, -34, -74, 66, 45, -85, -9, 89, -30,-77, 63, 50, -84, -13, 90, -26, -78, 60, 53, -82, -17, 90, -21, -80,56,}

{85, -53, -53, 85, 0, -85, 53, 53, -85, 0, 85, -53, -53, 85, 0, -85, 53,53, -85, 0, 85, -53, -53, 85, 0, -85, 53, 53, -85, 0, 85, -53,}

{82, -66, -30, 90, -42, -56, 86, -13, -77, 74, 17, -88, 53, 45, -89, 26,68, -80, -4, 84, -63, -34, 90, -38, -60, 85, -9, -78, 72, 21, -88, 50,}

{78, -77, -4, 80, -74, -9, 82, -72, -13, 84, -68, -17, 85, -66, -21, 86,-63, -26, 88, -60, -30, 88, -56, -34, 89, -53, -38, 90, -50, -42, 90,-45,}

{74, -84, 21, 60, -89, 42, 42, -89, 60, 21, -84, 74, 0, -74, 84, -21,-60, 89, -42, -42, 89, -60, -21, 84, -74, 0, 74, -84, 21, 60, -89, 42,}

{68, -88, 45, 30, -84, 78, -17, -56, 90, -60, -13, 77, -85, 34, 42, -88,72, -4, -66, 89, -50, -26, 82, -80, 21, 53, -90, 63, 9, -74, 86, -38,}

{63, -90, 66, -4, -60, 90, -68, 9, 56, -89, 72, -13, -53, 88, -74, 17,50, -88, 77, -21, -45, 86, -78, 26, 42, -85, 80, -30, -38, 84, -82, 34,}

{56, -88, 80, -38, -21, 72, -90, 68, -17, -42, 82, -86, 53, 4, -60, 88,-78, 34, 26, -74, 90, -66, 13, 45, -84, 85, -50, -9, 63, -89, 77, -30,}

{50, -82, 88, -66, 21, 30, -72, 90, -78, 42, 9, -56, 85, -86, 60, -13,-38, 77, -90, 74, -34, -17, 63, -88, 84, -53, 4, 45, -80, 89, -68, 26,}

{42, -74, 89, -84, 60, -21, -21, 60, -84, 89, -74, 42, 0, -42, 74, -89,84, -60, 21, 21, -60, 84, -89, 74, -42, 0, 42, -74, 89, -84, 60, -21,}

{34, -63, 82, -90, 84, -66, 38, -4, -30, 60, -80, 90, -85, 68, -42, 9,26, -56, 78, -89, 86, -72, 45, -13, -21, 53, -77, 88, -88, 74, -50, 17,}

{26, -50, 68, -82, 89, -88, 80, -66, 45, -21, -4, 30, -53, 72, -84, 90,-88, 78, -63, 42, -17, -9, 34, -56, 74, -85, 90, -86, 77, -60, 38, -13,}

{17, -34, 50, -63, 74, -82, 88, -90, 88, -84, 77, -66, 53, -38, 21, -4,-13, 30, -45, 60, -72, 80, -86, 90, -89, 85, -78, 68, -56, 42, -26, 9,}

{9, -17, 26, -34, 42, -50, 56, -63, 68, -74, 78, -82, 85, -88, 89, -90,90, -88, 86, -84, 80, -77, 72, -66, 60, -53, 45, -38, 30, -21, 13, -4,}

64-Bit DST-7:

{2, 5, 7, 8, 11, 13, 15, 17, 20, 22, 23, 26, 29, 30, 32, 34, 37, 38, 41,42, 44, 46, 48, 49, 52, 53, 55, 58, 59, 60, 62, 64, 65, 66, 68, 69, 72,72, 74, 73, 76, 77, 78, 79, 81, 80, 82, 83, 85, 84, 85, 86, 87, 87, 88,89, 89, 90, 89, 90, 90, 90, 90, 90,}

{7, 13, 20, 26, 32, 38, 44, 49, 55, 60, 65, 69, 74, 77, 81, 83, 85, 87,89, 90, 90, 90, 90, 89, 87, 85, 83, 81, 77, 74, 69, 65, 60, 55, 49, 44,38, 32, 26, 20, 13, 7, 0, -7, -13, -20, -26, -32, -38, -44, -49, -55,-60, -65, -69, -74, -77, -81, -83, -85, -87, -89, -90, -90,}{11, 22, 32, 42, 52, 60, 68, 73, 81, 84, 88, 90, 90, 89, 87, 85, 79, 74,66, 59, 49, 41, 30, 20, 8, -2, -13, -23, -34, -44, -53, -62, -69, -76,-80, -85, -89, -90, -90, -90, -87, -83, -78, -72, -65, -58, -48, -38,-29, -17, -7, 5, 15, 26, 37, 46, 55, 64, 72, 77, 82, 86, 89, 90,}{15, 30, 44, 58, 68, 77, 85, 89, 90, 89, 86, 81, 72, 62, 49, 37, 22, 7,-8, -23, -38, -52, -64, -74, -80, -87, -90, -90, -88, -83, -76, -66,-55, -42, -29, -13, 2, 17, 32, 46, 59, 69, 78, 84, 89, 90, 90, 85, 79,72, 60, 48, 34, 20, 5, -11, -26, -41, -53, -65, -73, -82, -87, -90,}{20, 38, 55, 69, 81, 87, 90, 89, 83, 74, 60, 44, 26, 7, -13, -32, -49,-65, -77, -85, -90, -90, -85, -77, -65, -49, -32, -13, 7, 26, 44, 60,74, 83, 89, 90, 87, 81, 69, 55, 38, 20, 0, -20, -38, -55, -69, -81, -87,-90, -89, -83, -74, -60, -44, -26, -7, 13, 32, 49, 65, 77, 85, 90,}{23, 46, 65, 79, 88, 90, 86, 76, 60, 41, 17, -7, -30, -52, -69, -82,-90, -90, -85, -72, -55, -34, -11, 13, 37, 58, 74, 84, 90, 89, 80, 68,49, 29, 5, -20, -42, -62, -77, -87, -90, -87, -78, -64, -44, -22, 2, 26,48, 66, 81, 89, 90, 85, 73, 59, 38, 15, -8, -32, -53, -72, -83, -89,}{29, 53, 74, 86, 90, 85, 72, 52, 26, -2, -30, -55, -73, -87, -90, -84,-72, -49, -23, 5, 32, 58, 76, 87, 90, 85, 69, 48, 22, -7, -34, -59, -77,-88, -90, -83, -68, -46, -20, 8, 37, 60, 78, 89, 90, 82, 66, 44, 17,-11, -38, -62, -79, -89, -89, -80, -65, -42, -15, 13, 41, 64, 81, 90,}{32, 60, 81, 90, 87, 74, 49, 20, -13, -44, -69, -85, -90, -83, -65, -38,-7, 26, 55, 77, 89, 89, 77, 55, 26, -7, -38, -65, -83, -90, -85, -69,-44, -13, 20, 49, 74, 87, 90, 81, 60, 32, 0, -32, -60, -81, -90, -87,-74, -49, -20, 13, 44, 69, 85, 90, 83, 65, 38, 7, -26, -55, -77, -89,}{37, 66, 85, 90, 79, 55, 22, -15, -49, -76, -90, -87, -72, -42, -7, 30,62, 83, 90, 82, 60, 29, -8, -44, -72, -88, -89, -73, -48, -13, 23, 58,81, 90, 84, 65, 34, -2, -38, -68, -86, -90, -78, -53, -20, 17, 52, 77,89, 87, 69, 41, 5, -32, -64, -85, -90, -80, -59, -26, 11, 46, 74, 89,}{41, 72, 89, 87, 66, 32, -8, -48, -77, -90, -85, -60, -23, 17, 55, 80,90, 81, 53, 15, -26, -62, -84, -90, -76, -46, -7, 34, 68, 87, 89, 72,38, -2, -42, -74, -90, -86, -65, -30, 11, 49, 78, 90, 83, 59, 22, -20,-58, -82, -90, -79, -52, -13, 29, 64, 85, 89, 73, 44, 5, -37, -69, -88,}{44, 77, 90, 81, 49, 7, -38, -74, -90, -83, -55, -13, 32, 69, 89, 85,60, 20, -26, -65, -87, -87, -65, -26, 20, 60, 85, 89, 69, 32, -13, -55,-83, -90, -74, -38, 7, 49, 81, 90, 77, 44, 0, -44, -77, -90, -81, -49,-7, 38, 74, 90, 83, 55, 13, -32, -69, -89, -85, -60, -20, 26, 65, 87,}{48, 80, 90, 72, 30, -20, -64, -88, -85, -58, -11, 38, 76, 90, 77, 41,-8, -55, -84, -89, -65, -22, 29, 69, 89, 82, 49, 2, -46, -81, -90, -72,-32, 17, 62, 87, 86, 59, 13, -37, -73, -90, -78, -42, 7, 53, 85, 89, 66,23, -26, -68, -90, -83, -52, -5, 44, 79, 90, 74, 34, -15, -60, -87,}{52, 84, 87, 59, 8, -44, -80, -90, -65, -17, 37, 77, 90, 72, 26, -29,-72, -90, -76, -34, 20, 66, 89, 81, 42, -11, -60, -88, -85, -49, 2, 53,85, 87, 58, 7, -46, -82, -89, -64, -15, 38, 78, 90, 69, 23, -30, -74,-90, -73, -32, 22, 68, 90, 79, 41, -13, -62, -89, -83, -48, 5, 55, 86,}{55, 87, 83, 44, -13, -65, -90, -77, -32, 26, 74, 90, 69, 20, -38, -81,-89, -60, -7, 49, 85, 85, 49, -7, -60, -89, -81, -38, 20, 69, 90, 74,26, -32, -77, -90, -65, -13, 44, 83, 87, 55, 0, -55, -87, -83, -44, 13,65, 90, 77, 32, -26, -74, -90, -69, -20, 38, 81, 89, 60, 7, -49, -85,}{59, 90, 77, 29, -34, -81, -88, -53, 7, 64, 90, 74, 22, -41, -83, -86,-48, 13, 68, 90, 69, 15, -46, -85, -85, -42, 20, 72, 90, 65, 8, -52,-87, -80, -37, 26, 76, 89, 60, 2, -58, -89, -78, -30, 32, 79, 89, 55,-5, -62, -90, -73, -23, 38, 82, 87, 49, -11, -66, -90, -72, -17, 44,84,}{62, 90, 69, 11, -53, -89, -76, -22, 44, 86, 80, 32, -34, -82, -85, -42,23, 77, 89, 52, -13, -72, -90, -60, 2, 64, 90, 68, 8, -55, -90, -73,-20, 46, 87, 81, 30, -37, -83, -84, -41, 26, 78, 88, 49, -15, -72, -90,-59, 5, 65, 90, 66, 7, -58, -89, -74, -17, 48, 87, 79, 29, -38, -85,}{65, 90, 60, -7, -69, -90, -55, 13, 74, 89, 49, -20, -77, -87, -44, 26,81, 85, 38, -32, -83, -83, -32, 38, 85, 81, 26, -44, -87, -77, -20, 49,89, 74, 13, -55, -90, -69, -7, 60, 90, 65, 0, -65, -90, -60, 7, 69, 90,55, -13, -74, -89, -49, 20, 77, 87, 44, -26, -81, -85, -38, 32, 83,}{68, 89, 49, -23, -80, -83, -29, 46, 89, 72, 5, -65, -90, -53, 20, 79,84, 32, -42, -88, -74, -8, 62, 90, 58, -15, -77, -86, -37, 38, 87, 76,13, -59, -90, -60, 11, 73, 87, 41, -34, -85, -78, -17, 55, 90, 64, -7,-72, -89, -44, 30, 85, 81, 22, -52, -90, -66, 2, 69, 90, 48, -26, -82,}{72, 88, 38, -41, -89, -69, 2, 72, 87, 37, -42, -89, -68, 5, 74, 87, 34,-44, -90, -66, 7, 73, 86, 32, -46, -89, -65, 8, 76, 85, 30, -48, -90,-64, 11, 77, 84, 29, -49, -90, -62, 13, 78, 85, 26, -52, -90, -60, 15,79, 83, 23, -53, -90, -59, 17, 81, 82, 22, -55, -90, -58, 20, 80,}{74, 85, 26, -55, -90, -49, 32, 87, 69, -7, -77, -83, -20, 60, 90, 44,-38, -89, -65, 13, 81, 81, 13, -65, -89, -38, 44, 90, 60, -20, -83, -77,-7, 69, 87, 32, -49, -90, -55, 26, 85, 74, 0, -74, -85, -26, 55, 90, 49,-32, -87, -69, 7, 77, 83, 20, -60, -90, -44, 38, 89, 65, -13, -81,}{76, 82, 13, -68, -87, -26, 59, 89, 38, -48, -90, -49, 37, 90, 60, -23,-86, -69, 11, 80, 77, 2, -73, -83, -15, 66, 87, 29, -58, -90, -41, 46,90, 52, -34, -89, -62, 22, 85, 72, -8, -81, -78, -5, 74, 85, 17, -65,-88, -30, 55, 90, 42, -44, -90, -53, 32, 89, 64, -20, -84, -72, 7, 79,}{78, 78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0,78, 78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0,78, 78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0,78, 78, 0, -78, -78, 0, 78, 78, 0, -78,}{81, 74, -13, -85, -65, 26, 89, 55, -38, -90, -44, 49, 90, 32, -60, -87,-20, 69, 83, 7, -77, -77, 7, 83, 69, -20, -87, -60, 32, 90, 49, -44,-90, -38, 55, 89, 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4-Point DCT-8:

{84, 74, 55, 29,}

{74, 0, -74, -74,}

{55, -74, -29, 84,}

{29, -74, 84, -55,}

8-Point DCT-8:

{86, 85, 78, 71, 60, 46, 32, 17,}

{85, 60, 17, -32, -71, -86, -78, -46,}

{78, 17, -60, -86, -46, 32, 85, 71,}

{71, -32, -86, -17, 78, 60, -46, -85,}

{60, -71, -46, 78, 32, -85, -17, 86,}

{46, -86, 32, 60, -85, 17, 71, -78,}

{32, -78, 85, -46, -17, 71, -86, 60,}

{17, -46, 71, -85, 86, -78, 60, -32,}

16-Point DCT-8:

{90, 89, 87, 83, 81, 77, 72, 66, 62, 56, 49, 41, 33, 25, 17, 9,}

{89, 81, 66, 49, 25, 0, -25, -49, -66, -81, -89, -89, -81, -66, -49,-25,}

{87, 66, 33, -9, -49, -77, -90, -81, -56, -17, 25, 62, 83, 89, 72, 41,}

{83, 49, -9, -62, -89, -77, -33, 25, 72, 90, 66, 17, -41, -81, -87,-56,}

{81, 25, -49, -89, -66, 0, 66, 89, 49, -25, -81, -81, -25, 49, 89, 66,}

{77, 0, -77, -77, 0, 77, 77, 0, -77, -77, 0, 77, 77, 0, -77, -77,}

{72, -25, -90, -33, 66, 77, -17, -89, -41, 62, 81, -9, -87, -49, 56,83,}

{66, -49, -81, 25, 89, 0, -89, -25, 81, 49, -66, -66, 49, 81, -25, -89,}

{62, -66, -56, 72, 49, -77, -41, 81, 33, -83, -25, 87, 17, -89, -9, 90,}

{56, -81, -17, 90, -25, -77, 62, 49, -83, -9, 89, -33, -72, 66, 41,-87,}

{49, -89, 25, 66, -81, 0, 81, -66, -25, 89, -49, -49, 89, -25, -66, 81,}

{41, -89, 62, 17, -81, 77, -9, -66, 87, -33, -49, 90, -56, -25, 83,-72,}

{33, -81, 83, -41, -25, 77, -87, 49, 17, -72, 89, -56, -9, 66, -90, 62,}

{25, -66, 89, -81, 49, 0, -49, 81, -89, 66, -25, -25, 66, -89, 81, -49,}

{17, -49, 72, -87, 89, -77, 56, -25, -9, 41, -66, 83, -90, 81, -62, 33,}

{9, -25, 41, -56, 66, -77, 83, -89, 90, -87, 81, -72, 62, -49, 33, -17,}

32-Point DCT-8:

{90, 90, 89, 88, 88, 86, 85, 84, 82, 80, 78, 77, 74, 72, 68, 66, 63, 60,56, 53, 50, 45, 42, 38, 34, 30, 26, 21, 17, 13, 9, 4,}

{90, 88, 84, 78, 72, 63, 53, 42, 30, 17, 4, -9, -21, -34, -45, -56, -66,-74, -80, -85, -88, -90, -89, -86, -82, -77, -68, -60, -50, -38, -26,-13,}

{89, 84, 74, 60, 42, 21, 0, -21, -42, -60, -74, -84, -89, -89, -84, -74,-60, -42, -21, 0, 21, 42, 60, 74, 84, 89, 89, 84, 74, 60, 42, 21,}

{88, 78, 60, 34, 4, -26, -53, -74, -86, -90, -82, -66, -42, -13, 17, 45,68, 84, 90, 85, 72, 50, 21, -9, -38, -63, -80, -89, -88, -77, -56, -30,}

{88, 72, 42, 4, -34, -66, -85, -89, -77, -50, -13, 26, 60, 82, 90, 80,56, 21, -17, -53, -78, -90, -84, -63, -30, 9, 45, 74, 88, 86, 68, 38,}

{86, 63, 21, -26, -66, -88, -85, -60, -17, 30, 68, 88, 84, 56, 13, -34,-72, -89, -82, -53, -9, 38, 74, 90, 80, 50, 4, -42, -77, -90, -78, -45,}

{85, 53, 0, -53, -85, -85, -53, 0, 53, 85, 85, 53, 0, -53, -85, -85,-53, 0, 53, 85, 85, 53, 0, -53, -85, -85, -53, 0, 53, 85, 85, 53,}

{84, 42, -21, -74, -89, -60, 0, 60, 89, 74, 21, -42, -84, -84, -42, 21,74, 89, 60, 0, -60, -89, -74, -21, 42, 84, 84, 42, -21, -74, -89, -60,}

{82, 30, -42, -86, -77, -17, 53, 89, 68, 4, -63, -90, -60, 9, 72, 88,50, -21, -78, -85, -38, 34, 84, 80, 26, -45, -88, -74, -13, 56, 90, 66,}

{80, 17, -60, -90, -50, 30, 85, 74, 4, -68, -88, -38, 42, 88, 66, -9,-77, -84, -26, 53, 90, 56, -21, -82, -78, -13, 63, 89, 45, -34, -86,-72,}

{78, 4, -74, -82, -13, 68, 85, 21, -63, -88, -30, 56, 89, 38, -50, -90,-45, 42, 90, 53, -34, -88, -60, 26, 86, 66, -17, -84, -72, 9, 80, 77,}

{77, -9, -84, -66, 26, 88, 53, -42, -90, -38, 56, 88, 21, -68, -82, -4,78, 74, -13, -85, -63, 30, 89, 50, -45, -90, -34, 60, 86, 17, -72, -80,}

{74, -21, -89, -42, 60, 84, 0, -84, -60, 42, 89, 21, -74, -74, 21, 89,42, -60, -84, 0, 84, 60, -42, -89, -21, 74, 74, -21, -89, -42, 60, 84,}

{72, -34, -89, -13, 82, 56, -53, -84, 9, 88, 38, -68, -74, 30, 90, 17,-80, -60, 50, 85, -4, -88, -42, 66, 77, -26, -90, -21, 78, 63, -45,-86,}

{68, -45, -84, 17, 90, 13, -85, -42, 72, 66, -50, -82, 21, 90, 9, -86,-38, 74, 63, -53, -80, 26, 89, 4, -88, -34, 77, 60, -56, -78, 30, 88,}

{66, -56, -74, 45, 80, -34, -85, 21, 88, -9, -90, -4, 89, 17, -86, -30,82, 42, -77, -53, 68, 63, -60, -72, 50, 78, -38, -84, 26, 88, -13, -90,}

{63, -66, -60, 68, 56, -72, -53, 74, 50, -77, -45, 78, 42, -80, -38, 82,34, -84, -30, 85, 26, -86, -21, 88, 17, -88, -13, 89, 9, -90, -4, 90,}

{60, -74, -42, 84, 21, -89, 0, 89, -21, -84, 42, 74, -60, -60, 74, 42,-84, -21, 89, 0, -89, 21, 84, -42, -74, 60, 60, -74, -42, 84, 21, -89,}

{56, -80, -21, 90, -17, -82, 53, 60, -78, -26, 90, -13, -84, 50, 63,-77, -30, 89, -9, -85, 45, 66, -74, -34, 88, -4, -86, 42, 68, -72, -38,88,}

{53, -85, 0, 85, -53, -53, 85, 0, -85, 53, 53, -85, 0, 85, -53, -53, 85,0, -85, 53, 53, -85, 0, 85, -53, -53, 85, 0, -85, 53, 53, -85,}

{50, -88, 21, 72, -78, -9, 85, -60, -38, 90, -34, -63, 84, -4, -80, 68,26, -89, 45, 53, -88, 17, 74, -77, -13, 86, -56, -42, 90, -30, -66, 82,}

{45, -90, 42, 50, -90, 38, 53, -89, 34, 56, -88, 30, 60, -88, 26, 63,-86, 21, 66, -85, 17, 68, -84, 13, 72, -82, 9, 74, -80, 4, 77, -78,}

{42, -89, 60, 21, -84, 74, 0, -74, 84, -21, -60, 89, -42, -42, 89, -60,-21, 84, -74, 0, 74, -84, 21, 60, -89, 42, 42, -89, 60, 21, -84, 74,}

{38, -86, 74, -9, -63, 90, -53, -21, 80, -82, 26, 50, -89, 66, 4, -72,88, -42, -34, 85, -77, 13, 60, -90, 56, 17, -78, 84, -30, -45, 88, -68,}

{34, -82, 84, -38, -30, 80, -85, 42, 26, -78, 86, -45, -21, 77, -88, 50,17, -74, 88, -53, -13, 72, -89, 56, 9, -68, 90, -60, -4, 66, -90, 63,}

{30, -77, 89, -63, 9, 50, -85, 84, -45, -13, 66, -90, 74, -26, -34, 78,-88, 60, -4, -53, 86, -82, 42, 17, -68, 90, -72, 21, 38, -80, 88, -56,}

{26, -68, 89, -80, 45, 4, -53, 84, -88, 63, -17, -34, 74, -90, 77, -38,-13, 60, -86, 85, -56, 9, 42, -78, 90, -72, 30, 21, -66, 88, -82, 50,}

{21, -60, 84, -89, 74, -42, 0, 42, -74, 89, -84, 60, -21, -21, 60, -84,89, -74, 42, 0, -42, 74, -89, 84, -60, 21, 21, -60, 84, -89, 74, -42,}

{17, -50, 74, -88, 88, -77, 53, -21, -13, 45, -72, 86, -89, 78, -56, 26,9, -42, 68, -85, 90, -80, 60, -30, -4, 38, -66, 84, -90, 82, -63, 34,}

{13, -38, 60, -77, 86, -90, 85, -74, 56, -34, 9, 17, -42, 63, -78, 88,-90, 84, -72, 53, -30, 4, 21, -45, 66, -80, 88, -89, 82, -68, 50, -26,}

{9, -26, 42, -56, 68, -78, 85, -89, 90, -86, 80, -72, 60, -45, 30, -13,-4, 21, -38, 53, -66, 77, -84, 88, -90, 88, -82, 74, -63, 50, -34, 17,}

{4, -13, 21, -30, 38, -45, 53, -60, 66, -72, 77, -80, 84, -86, 88, -90,90, -89, 88, -85, 82, -78, 74, -68, 63, -56, 50, -42, 34, -26, 17, -9,}

64-Point DCT-8:

{90, 90, 90, 90, 90, 89, 90, 89, 89, 88, 87, 87, 86, 85, 84, 85, 83, 82,80, 81, 79, 78, 77, 76, 73, 74, 72, 72, 69, 68, 66, 65, 64, 62, 60, 59,58, 55, 53, 52, 49, 48, 46, 44, 42, 41, 38, 37, 34, 32, 30, 29, 26, 23,22, 20, 17, 15, 13, 11, 8, 7, 5, 2,}

{90, 90, 89, 87, 85, 83, 81, 77, 74, 69, 65, 60, 55, 49, 44, 38, 32, 26,20, 13, 7, 0, -7, -13, -20, -26, -32, -38, -44, -49, -55, -60, -65, -69,-74, -77, -81, -83, -85, -87, -89, -90, -90, -90, -90, -89, - 87, -85,-83, -81, -77, -74, -69, -65, -60, -55, -49, -44, -38, -32, -26, -20,-13, -7,}{90, 89, 86, 82, 77, 72, 64, 55, 46, 37, 26, 15, 5, -7, -17, -29, -38,-48, -58, -65, -72, -78, -83, -87, -90, -90, -90, -89, -85, -80, -76,-69, -62, -53, -44, -34, -23, -13, -2, 8, 20, 30, 41, 49, 59, 66, 74,79, 85, 87, 89, 90, 90, 88, 84, 81, 73, 68, 60, 52, 42, 32, 22, 11,}{90, 87, 82, 73, 65, 53, 41, 26, 11, -5, -20, -34, -48, -60, -72, -79,-85, -90, -90, -89, -84, -78, -69, -59, -46, -32, -17, -2, 13, 29, 42,55, 66, 76, 83, 88, 90, 90, 87, 80, 74, 64, 52, 38, 23, 8, -7, -22, -37,-49, -62, -72, -81, -86, -89, -90, -89, -85, -77, -68, -58, -44, -30,-15,}{90, 85, 77, 65, 49, 32, 13, -7, -26, -44, -60, -74, -83, -89, -90, -87,-81, -69, -55, -38, -20, 0, 20, 38, 55, 69, 81, 87, 90, 89, 83, 74, 60,44, 26, 7, -13, -32, -49, -65, -77, -85, -90, -90, -85, -77, -65, -49,-32, -13, 7, 26, 44, 60, 74, 83, 89, 90, 87, 81, 69, 55, 38, 20,}{89, 83, 72, 53, 32, 8, -15, -38, -59, -73, -85, -90, -89, -81, -66,-48, -26, -2, 22, 44, 64, 78, 87, 90, 87, 77, 62, 42, 20, -5, -29, -49,-68, -80, -89, -90, -84, -74, -58, -37, -13, 11, 34, 55, 72, 85, 90, 90,82, 69, 52, 30, 7, -17, -41, -60, -76, -86, -90, -88, -79, -65, -46,-23,}{90, 81, 64, 41, 13, -15, -42, -65, -80, -89, -89, -79, -62, -38, -11,17, 44, 66, 82, 90, 89, 78, 60, 37, 8, -20, -46, -68, -83, -90, -88,-77, -59, -34, -7, 22, 48, 69, 85, 90, 87, 76, 58, 32, 5, -23, -49, -72,-84, -90, -87, -73, -55, -30, -2, 26, 52, 72, 85, 90, 86, 74, 53, 29,}{89, 77, 55, 26, -7, -38, -65, -83, -90, -85, -69, -44, -13, 20, 49, 74,87, 90, 81, 60, 32, 0, -32, -60, -81, -90, -87, -74, -49, -20, 13, 44,69, 85, 90, 83, 65, 38, 7, -26, -55, -77, -89, -89, -77, -55, -26, 7,38, 65, 83, 90, 85, 69, 44, 13, -20, -49, -74, -87, -90, -81, -60, -32,}{89, 74, 46, 11, -26, -59, -80, -90, -85, -64, -32, 5, 41, 69, 87, 89,77, 52, 17, -20, -53, -78, -90, -86, -68, -38, -2, 34, 65, 84, 90, 81,58, 23, -13, -48, -73, -89, -88, -72, -44, -8, 29, 60, 82, 90, 83, 62,30, -7, -42, -72, -87, -90, -76, -49, -15, 22, 55, 79, 90, 85, 66, 37,}{88, 69, 37, -5, -44, -73, -89, -85, -64, -29, 13, 52, 79, 90, 82, 58,20, -22, -59, -83, -90, -78, -49, -11, 30, 65, 86, 90, 74, 42, 2, -38,-72, -89, -87, -68, -34, 7, 46, 76, 90, 84, 62, 26, -15, -53, -81, -90,-80, -55, -17, 23, 60, 85, 90, 77, 48, 8, -32, -66, -87, -89, -72, -41,}{87, 65, 26, -20, -60, -85, -89, -69, -32, 13, 55, 83, 90, 74, 38, -7,-49, -81, -90, -77, -44, 0, 44, 77, 90, 81, 49, 7, -38, -74, -90, -83,-55, -13, 32, 69, 89, 85, 60, 20, -26, -65, -87, -87, -65, -26, 20, 60,85, 89, 69, 32, -13, -55, -83, -90, -74, -38, 7, 49, 81, 90, 77, 44,}{87, 60, 15, -34, -74, -90, -79, -44, 5, 52, 83, 90, 68, 26, -23, -66,-89, -85, -53, -7, 42, 78, 90, 73, 37, -13, -59, -86, -87, -62, -17, 32,72, 90, 81, 46, -2, -49, -82, -89, -69, -29, 22, 65, 89, 84, 55, 8, -41,-77, -90, -76, -38, 11, 58, 85, 88, 64, 20, -30, -72, -90, -80, -48,}{86, 55, 5, -48, -83, -89, -62, -13, 41, 79, 90, 68, 22, -32, -73, -90,-74, -30, 23, 69, 90, 78, 38, -15, -64, -89, -82, -46, 7, 58, 87, 85,53, 2, -49, -85, -88, -60, -11, 42, 81, 89, 66, 20, -34, -76, -90, -72,-29, 26, 72, 90, 77, 37, -17, -65, -90, -80, -44, 8, 59, 87, 84, 52,}{85, 49, -7, -60, -89, -81, -38, 20, 69, 90, 74, 26, -32, -77, -90, -65,-13, 44, 83, 87, 55, 0, -55, -87, -83, -44, 13, 65, 90, 77, 32, -26,-74, -90, -69, -20, 38, 81, 89, 60, 7, -49, -85, -85, -49, 7, 60, 89,81, 38, -20, -69, -90, -74, -26, 32, 77, 90, 65, 13, -44, -83, -87,-55,}{84, 44, -17, -72, -90, -66, -11, 49, 87, 82, 38, -23, -73, -90, -62,-5, 55, 89, 79, 32, -30, -78, -89, -58, 2, 60, 89, 76, 26, -37, -80,-87, -52, 8, 65, 90, 72, 20, -42, -85, -85, -46, 15, 69, 90, 68, 13,-48, -86, -83, -41, 22, 74, 90, 64, 7, -53, -88, -81, -34, 29, 77, 90,59,}{85, 38, -29, -79, -87, -48, 17, 74, 89, 58, -7, -66, -90, -65, -5, 59,90, 72, 15, -49, -88, -78, -26, 41, 84, 83, 37, -30, -81, -87, -46, 20,73, 90, 55, -8, -68, -90, -64, -2, 60, 90, 72, 13, -52, -89, -77, -23,42, 85, 82, 34, -32, -80, -86, -44, 22, 76, 89, 53, -11, -69, -90, -62,}{83, 32, -38, -85, -81, -26, 44, 87, 77, 20, -49, -89, -74, -13, 55, 90,69, 7, -60, -90, -65, 0, 65, 90, 60, -7, -69, -90, -55, 13, 74, 89, 49,-20, -77, -87, -44, 26, 81, 85, 38, -32, -83, -83, -32, 38, 85, 81, 26,-44, -87, -77, -20, 49, 89, 74, 13, -55, -90, -69, -7, 60, 90, 65,}{82, 26, -48, -90, -69, -2, 66, 90, 52, -22, -81, -85, -30, 44, 89, 72,7, -64, -90, -55, 17, 78, 85, 34, -41, -87, -73, -11, 60, 90, 59, -13,-76, -87, -38, 37, 86, 77, 15, -58, -90, -62, 8, 74, 88, 42, -32, -84,-79, -20, 53, 90, 65, -5, -72, -89, -46, 29, 83, 80, 23, -49, -89, -68,}{80, 20, -58, -90, -55, 22, 82, 81, 17, -59, -90, -53, 23, 83, 79, 15,-60, -90, -52, 26, 85, 78, 13, -62, -90, -49, 29, 84, 77, 11, -64, -90,-48, 30, 85, 76, 8, -65, -89, -46, 32, 86, 73, 7, -66, -90, -44, 34, 87,74, 5, -68, -89, -42, 37, 87, 72, 2, -69, -89, -41, 38, 88, 72,}{81, 13, -65, -89, -38, 44, 90, 60, -20, -83, -77, -7, 69, 87, 32, -49,-90, -55, 26, 85, 74, 0, -74, -85, -26, 55, 90, 49, -32, -87, -69, 7,77, 83, 20, -60, -90, -44, 38, 89, 65, -13, -81, -81, -13, 65, 89, 38,-44, -90, -60, 20, 83, 77, 7, -69, -87, -32, 49, 90, 55, -26, -85, -74,}{79, 7, -72, -84, -20, 64, 89, 32, -53, -90, -44, 42, 90, 55, -30, -88,-65, 17, 85, 74, -5, -78, -81, -8, 72, 85, 22, -62, -89, -34, 52, 90,46, -41, -90, -58, 29, 87, 66, -15, -83, -73, 2, 77, 80, 11, -69, -86,-23, 60, 90, 37, -49, -90, -48, 38, 89, 59, -26, -87, -68, 13, 82, 76,}{78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0, 78,78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0, 78,78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0, 78, 78, 0, -78, -78, 0, 78,78, 0, -78, -78, 0, 78, 78, 0, -78, -78,}{77, -7, -83, -69, 20, 87, 60, -32, -90, -49, 44, 90, 38, -55, -89, -26,65, 85, 13, -74, -81, 0, 81, 74, -13, -85, -65, 26, 89, 55, -38, -90,-44, 49, 90, 32, -60, -87, -20, 69, 83, 7, -77, -77, 7, 83, 69, -20,-87, -60, 32, 90, 49, -44, -90, -38, 55, 89, 26, -65, -85, -13, 74, 81,}{76, -13, -87, -59, 38, 90, 37, -60, -86, -11, 77, 73, -15, -87, -58,41, 90, 34, -62, -85, -8, 78, 74, -17, -88, -55, 42, 90, 32, -64, -84,-7, 79, 72, -20, -89, -53, 44, 90, 30, -65, -85, -5, 81, 72, -22, -89,-52, 46, 90, 29, -66, -83, -2, 80, 69, -23, -90, -49, 48, 89, 26, -68,-82,}{73, -20, -90, -46, 55, 87, 8, -81, -68, 30, 90, 37, -64, -83, 2, 84,60, -41, -90, -26, 72, 78, -13, -88, -52, 49, 89, 15, -77, -72, 23, 90,42, -59, -85, -5, 82, 65, -34, -90, -32, 66, 80, -7, -86, -58, 44, 89,22, -74, -76, 17, 89, 48, -53, -87, -11, 79, 69, -29, -90, -38, 62, 85,}{74, -26, -90, -32, 69, 77, -20, -90, -38, 65, 81, -13, -89, -44, 60,83, -7, -87, -49, 55, 85, 0, -85, -55, 49, 87, 7, -83, -60, 44, 89, 13,-81, -65, 38, 90, 20, -77, -69, 32, 90, 26, -74, -74, 26, 90, 32, -69,-77, 20, 90, 38, -65, -81, 13, 89, 44, -60, -83, 7, 87, 49, -55, -85,}{72, -32, -90, -17, 81, 62, -46, -87, -2, 86, 49, -59, -82, 13, 89, 37,-69, -73, 29, 90, 22, -78, -65, 42, 89, 7, -84, -53, 55, 85, -8, -89,-41, 66, 77, -23, -90, -26, 76, 68, -38, -90, -11, 83, 58, -52, -85, 5,88, 44, -64, -79, 20, 90, 30, -74, -72, 34, 90, 15, -80, -60, 48, 87,}{72, -38, -89, -2, 87, 42, -68, -74, 34, 90, 7, -86, -46, 65, 76, -30,-90, -11, 84, 49, -62, -78, 26, 90, 15, -83, -53, 59, 81, -22, -90, -20,80, 58, -55, -82, 17, 90, 23, -79, -60, 52, 85, -13, -90, -29, 77, 64,-48, -85, 8, 89, 32, -73, -66, 44, 87, -5, -89, -37, 72, 69, -41, -88,}{69, -44, -85, 13, 90, 20, -83, -49, 65, 74, -38, -87, 7, 90, 26, -81,-55, 60, 77, -32, -89, 0, 89, 32, -77, -60, 55, 81, -26, -90, -7, 87,38, -74, -65, 49, 83, -20, -90, -13, 85, 44, -69, -69, 44, 85, -13, -90,-20, 83, 49, -65, -74, 38, 87, -7, -90, -26, 81, 55, -60, -77, 32, 89,}{68, -49, -80, 29, 89, -5, -90, -20, 84, 42, -74, -62, 58, 77, -37, -87,13, 90, 11, -87, -34, 78, 55, -64, -72, 44, 85, -22, -90, -2, 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89, -48, -23,81, -84, 34, 38, -86, 78, -20, -52, 89, -69, 5, 64, -90, 59, 11, -74,89, -46, -26, 80, -85, 32, 41, -87, 77, -17, -53, 90, -68, 2, 65, -90,58, 13, -73, 88, -44, -29, 82, -83, 30, 42, -87, 76, -15, -55, 90, -66,}{34, -83, 85, -37, -32, 82, -84, 38, 30, -80, 85, -41, -29, 81, -86, 42,26, -79, 87, -44, -23, 78, -87, 46, 22, -77, 88, -48, -20, 76, -89, 49,17, -73, 89, -52, -15, 74, -90, 53, 13, -72, 89, -55, -11, 72, -90, 58,8, -69, 90, -59, -7, 68, -90, 60, 5, -66, 90, -62, -2, 65, -90, 64,}{32, -81, 87, -49, -13, 69, -90, 65, -7, -55, 89, -77, 26, 38, -83, 85,-44, -20, 74, -90, 60, 0, -60, 90, -74, 20, 44, -85, 83, -38, -26, 77,-89, 55, 7, -65, 90, -69, 13, 49, -87, 81, -32, -32, 81, -87, 49, 13,-69, 90, -65, 7, 55, -89, 77, -26, -38, 83, -85, 44, 20, -74, 90, -60,}{30, -77, 89, -62, 7, 52, -87, 83, -42, -17, 69, -90, 72, -20, -41, 82,-87, 53, 5, -60, 90, -78, 32, 29, -76, 90, -64, 8, 49, -86, 85, -44,-15, 68, -90, 72, -22, -38, 80, -88, 55, 2, -59, 89, -79, 34, 26, -73,90, 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37, -62, 81, -89, 89, -77, 58, -30,}{13, -38, 60, -77, 87, -90, 85, -74, 55, -32, 7, 20, -44, 65, -81, 89,-90, 83, -69, 49, -26, 0, 26, -49, 69, -83, 90, -89, 81, -65, 44, -20,-7, 32, -55, 74, -85, 90, -87, 77, -60, 38, -13, -13, 38, -60, 77, -87,90, -85, 74, -55, 32, -7, -20, 44, -65, 81, -89, 90, -83, 69, -49, 26,}{11, -32, 52, -68, 81, -88, 90, -87, 79, -66, 49, -30, 8, 13, -34, 53,-69, 80, -89, 90, -87, 78, -65, 48, -29, 7, 15, -37, 55, -72, 82, -89,90, -86, 77, -64, 46, -26, 5, 17, -38, 58, -72, 83, -90, 90, -85, 76,-62, 44, -23, 2, 20, -41, 59, -74, 85, -89, 90, -84, 73, -60, 42, -22,}{8, -26, 42, -58, 69, -79, 86, -90, 90, -87, 81, -72, 59, -44, 29, -11,-7, 23, -41, 55, -68, 78, -85, 89, -90, 87, -80, 72, -60, 46, -30, 13,5, -22, 38, -53, 66, -77, 84, -90, 90, -88, 82, -74, 62, -48, 32, -15,-2, 20, -37, 52, -65, 76, -85, 89, -90, 89, -83, 73, -64, 49, -34, 17,}{7, -20, 32, -44, 55, -65, 74, -81, 85, -89, 90, -90, 87, -83, 77, -69,60, -49, 38, -26, 13, 0, -13, 26, -38, 49, -60, 69, -77, 83, -87, 90,-90, 89, -85, 81, -74, 65, -55, 44, -32, 20, -7, -7, 20, -32, 44, -55,65, -74, 81, -85, 89, -90, 90, -87, 83, -77, 69, -60, 49, -38, 26, -13,}{5, -13, 22, -30, 38, -46, 53, -60, 66, -72, 77, -80, 84, -87, 90, -90,90, -89, 88, -85, 82, -78, 74, -68, 62, -55, 48, -41, 32, -23, 15, -7,-2, 11, -20, 29, -37, 44, -52, 59, -65, 72, -76, 81, -85, 87, -89, 90,-90, 90, -89, 86, -83, 79, -73, 69, -64, 58, -49, 42, -34, 26, -17, 8,}{2, -7, 11, -15, 20, -23, 29, -32, 37, -41, 44, -48, 52, -55, 59, -62,65, -68, 72, -74, 76, -78, 81, -82, 85, -85, 87, -88, 89, -89, 90, -90,90, -90, 90, -90, 89, -87, 86, -84, 83, -80, 79, -77, 73, -72, 69, -66,64, -60, 58, -53, 49, -46, 42, -38, 34, -30, 26, -22, 17, -13, 8, -5,}

After minimizing the cost value by trying all possible combination ofoffset values on a selected set of M unique numbers, T2 may be output asthe B-bit N-point primary transform core.

When deriving the 2N-Point DST-1 transform cores, it is restricted thatthe set of M unique numbers which construct the 2N-Point DST-1 corematrix is the same set of M unique numbers which construct the N-PointDST-7 and/or N-Point DCT-8 core matrix. In one embodiment, M equals toN. In one embodiment, one of the 8-Bit DST-1 transform cores shown belowmay be used. For example, in 8-Point DST-1, there are 4 unique numbersincluding 28, 55, 74, 84, and this is the same set of 4 unique numbersused to construct 4-Point DST-7 and 4-point DCT-8 cores using theembodiment where M equals N discussed above.

The 8-Bit DST-1 transform cores may include:

4-Point DST-1:

{48, 77, 77, 48,}

{77, 48, -48, -77,}

{77, -48, -48, 77,}

{48, -77, 77, -48,}

8-Point DST-1:

{29, 55, 74, 84, 84, 74, 55, 29,}

{55, 84, 74, 29, -29, -74, -84, -55,}

{74, 74, 0, -74, -74, 0, 74, 74,}

{84, 29, -74, -55, 55, 74, -29, -84,}

{84, -29, -74, 55, 55, -74, -29, 84,}

{74, -74, 0, 74, -74, 0, 74, -74,}

{55, -84, 74, -29, -29, 74, -84, 55,}

{29, -55, 74, -84, 84, -74, 55, -29,}

16-Point DST-1:

{17, 32, 46, 60, 71, 78, 85, 86, 86, 85, 78, 71, 60, 46, 32, 17,}

{32, 60, 78, 86, 85, 71, 46, 17, -17, -46, -71, -85, -86, -78, -60,-32,}

{46, 78, 86, 71, 32, -17, -60, -85, -85, -60, -17, 32, 71, 86, 78, 46,}

{60, 86, 71, 17, -46, -85, -78, -32, 32, 78, 85, 46, -17, -71, -86,-60,}

{71, 85, 32, -46, -86, -60, 17, 78, 78, 17, -60, -86, -46, 32, 85, 71,}

{78, 71, -17, -85, -60, 32, 86, 46, -46, -86, -32, 60, 85, 17, -71,-78,}

{85, 46, -60, -78, 17, 86, 32, -71, -71, 32, 86, 17, -78, -60, 46, 85,}

{86, 17, -85, -32, 78, 46, -71, -60, 60, 71, -46, -78, 32, 85, -17,-86,}

{86, -17, -85, 32, 78, -46, -71, 60, 60, -71, -46, 78, 32, -85, -17,86,}

{85, -46, -60, 78, 17, -86, 32, 71, -71, -32, 86, -17, -78, 60, 46,-85,}

{78, -71, -17, 85, -60, -32, 86, -46, -46, 86, -32, -60, 85, -17, -71,78,}

{71, -85, 32, 46, -86, 60, 17, -78, 78, -17, -60, 86, -46, -32, 85,-71,}

{60, -86, 71, -17, -46, 85, -78, 32, 32, -78, 85, -46, -17, 71, -86,60,}

{46, -78, 86, -71, 32, 17, -60, 85, -85, 60, -17, -32, 71, -86, 78,-46,}

{32, -60, 78, -86, 85, -71, 46, -17, -17, 46, -71, 85, -86, 78, -60,32,}

{17, -32, 46, -60, 71, -78, 85, -86, 86, -85, 78, -71, 60, -46, 32,-17,}

32-Point DST-1:

{9, 17, 25, 33, 41, 49, 56, 62, 66, 72, 77, 81, 83, 87, 89, 90, 90, 89,87, 83, 81, 77, 72, 66, 62, 56, 49, 41, 33, 25, 17, 9,}

{17, 33, 49, 62, 72, 81, 87, 90, 89, 83, 77, 66, 56, 41, 25, 9, -9, -25,-41, -56, -66, -77, -83, -89, -90, -87, -81, -72, -62, -49, -33, -17,}

{25, 49, 66, 81, 89, 89, 81, 66, 49, 25, 0, -25, -49, -66, -81, -89,-89, -81, -66, -49, -25, 0, 25, 49, 66, 81, 89, 89, 81, 66, 49, 25,}

{33, 62, 81, 90, 83, 66, 41, 9, -25, -56, -77, -89, -87, -72, -49, -17,17, 49, 72, 87, 89, 77, 56, 25, -9, -41, -66, -83, -90, -81, -62, -33,}

{41, 72, 89, 83, 62, 25, -17, -56, -81, -90, -77, -49, -9, 33, 66, 87,87, 66, 33, -9, -49, -77, -90, -81, -56, -17, 25, 62, 83, 89, 72, 41,}

{49, 81, 89, 66, 25, -25, -66, -89, -81, -49, 0, 49, 81, 89, 66, 25,-25, -66, -89, -81, -49, 0, 49, 81, 89, 66, 25, -25, -66, -89, -81,-49,}

{56, 87, 81, 41, -17, -66, -90, -72, -25, 33, 77, 89, 62, 9, -49, -83,-83, -49, 9, 62, 89, 77, 33, -25, -72, -90, -66, -17, 41, 81, 87, 56,}

{62, 90, 66, 9, -56, -89, -72, -17, 49, 87, 77, 25, -41, -83, -81, -33,33, 81, 83, 41, -25, -77, -87, -49, 17, 72, 89, 56, -9, -66, -90, -62,}

{66, 89, 49, -25, -81, -81, -25, 49, 89, 66, 0, -66, -89, -49, 25, 81,81, 25, -49, -89, -66, 0, 66, 89, 49, -25, -81, -81, -25, 49, 89, 66,}

{72, 83, 25, -56, -90, -49, 33, 87, 66, -9, -77, -81, -17, 62, 89, 41,-41, -89, -62, 17, 81, 77, 9, -66, -87, -33, 49, 90, 56, -25, -83, -72,}

{77, 77, 0, -77, -77, 0, 77, 77, 0, -77, -77, 0, 77, 77, 0, -77, -77, 0,77, 77, 0, -77, -77, 0, 77, 77, 0, -77, -77, 0, 77, 77,}

{81, 66, -25, -89, -49, 49, 89, 25, -66, -81, 0, 81, 66, -25, -89, -49,49, 89, 25, -66, -81, 0, 81, 66, -25, -89, -49, 49, 89, 25, -66, -81,}

{83, 56, -49, -87, -9, 81, 62, -41, -89, -17, 77, 66, -33, -90, -25, 72,72, -25, -90, -33, 66, 77, -17, -89, -41, 62, 81, -9, -87, -49, 56, 83,}

{87, 41, -66, -72, 33, 89, 9, -83, -49, 62, 77, -25, -90, -17, 81, 56,-56, -81, 17, 90, 25, -77, -62, 49, 83, -9, -89, -33, 72, 66, -41, -87,}

{89, 25, -81, -49, 66, 66, -49, -81, 25, 89, 0, -89, -25, 81, 49, -66,-66, 49, 81, -25, -89, 0, 89, 25, -81, -49, 66, 66, -49, -81, 25, 89,}

{90, 9, -89, -17, 87, 25, -83, -33, 81, 41, -77, -49, 72, 56, -66, -62,62, 66, -56, -72, 49, 77, -41, -81, 33, 83, -25, -87, 17, 89, -9, -90,}

{90, -9, -89, 17, 87, -25, -83, 33, 81, -41, -77, 49, 72, -56, -66, 62,62, -66, -56, 72, 49, -77, -41, 81, 33, -83, -25, 87, 17, -89, -9, 90,}

{89, -25, -81, 49, 66, -66, -49, 81, 25, -89, 0, 89, -25, -81, 49, 66,-66, -49, 81, 25, -89, 0, 89, -25, -81, 49, 66, -66, -49, 81, 25, -89,}

{87, -41, -66, 72, 33, -89, 9, 83, -49, -62, 77, 25, -90, 17, 81, -56,-56, 81, 17, -90, 25, 77, -62, -49, 83, 9, -89, 33, 72, -66, -41, 87,}

{83, -56, -49, 87, -9, -81, 62, 41, -89, 17, 77, -66, -33, 90, -25, -72,72, 25, -90, 33, 66, -77, -17, 89, -41, -62, 81, 9, -87, 49, 56, -83,}

{81, -66, -25, 89, -49, -49, 89, -25, -66, 81, 0, -81, 66, 25, -89, 49,49, -89, 25, 66, -81, 0, 81, -66, -25, 89, -49, -49, 89, -25, -66, 81,}

{77, -77, 0, 77, -77, 0, 77, -77, 0, 77, -77, 0, 77, -77, 0, 77, -77, 0,77, -77, 0, 77, -77, 0, 77, -77, 0, 77, -77, 0, 77, -77,}

{72, -83, 25, 56, -90, 49, 33, -87, 66, 9, -77, 81, -17, -62, 89, -41,-41, 89, -62, -17, 81, -77, 9, 66, -87, 33, 49, -90, 56, 25, -83, 72,}

{66, -89, 49, 25, -81, 81, -25, -49, 89, -66, 0, 66, -89, 49, 25, -81,81, -25, -49, 89, -66, 0, 66, -89, 49, 25, -81, 81, -25, -49, 89, -66,}

{62, -90, 66, -9, -56, 89, -72, 17, 49, -87, 77, -25, -41, 83, -81, 33,33, -81, 83, -41, -25, 77, -87, 49, 17, -72, 89, -56, -9, 66, -90, 62,}

{56, -87, 81, -41, -17, 66, -90, 72, -25, -33, 77, -89, 62, -9, -49, 83,-83, 49, 9, -62, 89, -77, 33, 25, -72, 90, -66, 17, 41, -81, 87, -56,}

{49, -81, 89, -66, 25, 25, -66, 89, -81, 49, 0, -49, 81, -89, 66, -25,-25, 66, -89, 81, -49, 0, 49, -81, 89, -66, 25, 25, -66, 89, -81, 49,}

{41, -72, 89, -83, 62, -25, -17, 56, -81, 90, -77, 49, -9, -33, 66, -87,87, -66, 33, 9, -49, 77, -90, 81, -56, 17, 25, -62, 83, -89, 72, -41,}

{33, -62, 81, -90, 83, -66, 41, -9, -25, 56, -77, 89, -87, 72, -49, 17,17, -49, 72, -87, 89, -77, 56, -25, -9, 41, -66, 83, -90, 81, -62, 33,}

{25, -49, 66, -81, 89, -89, 81, -66, 49, -25, 0, 25, -49, 66, -81, 89,-89, 81, -66, 49, -25, 0, 25, -49, 66, -81, 89, -89, 81, -66, 49, -25,}

{17, -33, 49, -62, 72, -81, 87, -90, 89, -83, 77, -66, 56, -41, 25, -9,-9, 25, -41, 56, -66, 77, -83, 89, -90, 87, -81, 72, -62, 49, -33, 17,}

{9, -17, 25, -33, 41, -49, 56, -62, 66, -72, 77, -81, 83, -87, 89, -90,90, -89, 87, -83, 81, -77, 72, -66, 62, -56, 49, -41, 33, -25, 17, -9,}

64-Point DST-1:

{4, 9, 13, 17, 21, 26, 30, 34, 38, 42, 45, 50, 53, 56, 60, 63, 66, 68,72, 74, 77, 78, 80, 82, 84, 85, 86, 88, 88, 89, 90, 90, 90, 90, 89, 88,88, 86, 85, 84, 82, 80, 78, 77, 74, 72, 68, 66, 63, 60, 56, 53, 50, 45,42, 38, 34, 30, 26, 21, 17, 13, 9, 4,}

{9, 17, 26, 34, 42, 50, 56, 63, 68, 74, 78, 82, 85, 88, 89, 90, 90, 88,86, 84, 80, 77, 72, 66, 60, 53, 45, 38, 30, 21, 13, 4, -4, -13, -21,-30, -38, -45, -53, -60, -66, -72, -77, -80, -84, -86, -88, -90, -90,-89, - 88, -85, -82, -78, -74, -68, -63, -56, -50, -42, -34, -26, -17,-9,}{13, 26, 38, 50, 60, 68, 77, 82, 86, 89, 90, 88, 85, 80, 74, 66, 56, 45,34, 21, 9, -4, -17, -30, -42, -53, -63, -72, -78, -84, -88, -90, -90,-88, -84, -78, -72, -63, -53, -42, -30, -17, -4, 9, 21, 34, 45, 56, 66,74, 80, 85, 88, 90, 89, 86, 82, 77, 68, 60, 50, 38, 26, 13,}{17, 34, 50, 63, 74, 82, 88, 90, 88, 84, 77, 66, 53, 38, 21, 4, -13,-30, -45, -60, -72, -80, -86, -90, -89, -85, -78, -68, -56, -42, -26,-9, 9, 26, 42, 56, 68, 78, 85, 89, 90, 86, 80, 72, 60, 45, 30, 13, -4,-21, -38, -53, -66, -77, -84, -88, -90, -88, -82, -74, -63, -50, -34,-17,}{21, 42, 60, 74, 84, 89, 89, 84, 74, 60, 42, 21, 0, -21, -42, -60, -74,-84, -89, -89, -84, -74, -60, -42, -21, 0, 21, 42, 60, 74, 84, 89, 89,84, 74, 60, 42, 21, 0, -21, -42, -60, -74, -84, -89, -89, -84, -74, -60,-42, -21, 0, 21, 42, 60, 74, 84, 89, 89, 84, 74, 60, 42, 21,}{26, 50, 68, 82, 89, 88, 80, 66, 45, 21, -4, -30, -53, -72, -84, -90,-88, -78, -63, -42, -17, 9, 34, 56, 74, 85, 90, 86, 77, 60, 38, 13, -13,-38, -60, -77, -86, -90, -85, -74, -56, -34, -9, 17, 42, 63, 78, 88, 90,84, 72, 53, 30, 4, -21, -45, -66, -80, -88, -89, -82, -68, -50, -26,}{30, 56, 77, 88, 89, 80, 63, 38, 9, -21, -50, -72, -85, -90, -84, -68,-45, -17, 13, 42, 66, 82, 90, 86, 74, 53, 26, -4, -34, -60, -78, -88,-88, -78, -60, -34, -4, 26, 53, 74, 86, 90, 82, 66, 42, 13, -17, -45,-68, -84, -90, -85, -72, -50, -21, 9, 38, 63, 80, 89, 88, 77, 56, 30,}{34, 63, 82, 90, 84, 66, 38, 4, -30, -60, -80, -90, -85, -68, -42, -9,26, 56, 78, 89, 86, 72, 45, 13, -21, -53, -77, -88, -88, -74, -50, -17,17, 50, 74, 88, 88, 77, 53, 21, -13, -45, -72, -86, -89, -78, -56, -26,9, 42, 68, 85, 90, 80, 60, 30, -4, -38, -66, -84, -90, -82, -63, -34,}{38, 68, 86, 88, 74, 45, 9, -30, -63, -84, -90, -78, -53, -17, 21, 56,80, 90, 82, 60, 26, -13, -50, -77, -89, -85, -66, -34, 4, 42, 72, 88,88, 72, 42, 4, -34, -66, -85, -89, -77, -50, -13, 26, 60, 82, 90, 80,56, 21, -17, -53, -78, -90, -84, -63, -30, 9, 45, 74, 88, 86, 68, 38,}{42, 74, 89, 84, 60, 21, -21, -60, -84, -89, -74, -42, 0, 42, 74, 89,84, 60, 21, -21, -60, -84, -89, -74, -42, 0, 42, 74, 89, 84, 60, 21,-21, -60, -84, -89, -74, -42, 0, 42, 74, 89, 84, 60, 21, -21, -60, -84,-89, -74, -42, 0, 42, 74, 89, 84, 60, 21, -21, -60, -84, -89, -74, -42,}{45, 78, 90, 77, 42, -4, -50, -80, -90, -74, -38, 9, 53, 82, 89, 72, 34,-13, -56, -84, -88, -68, -30, 17, 60, 85, 88, 66, 26, -21, -63, -86,-86, -63, -21, 26, 66, 88, 85, 60, 17, -30, -68, -88, -84, -56, -13, 34,72, 89, 82, 53, 9, -38, -74, -90, -80, -50, -4, 42, 77, 90, 78, 45,}{50, 82, 88, 66, 21, -30, -72, -90, -78, -42, 9, 56, 85, 86, 60, 13,-38, -77, -90, -74, -34, 17, 63, 88, 84, 53, 4, -45, -80, -89, -68, -26,26, 68, 89, 80, 45, -4, -53, -84, -88, -63, -17, 34, 74, 90, 77, 38,-13, -60, -86, -85, -56, -9, 42, 78, 90, 72, 30, -21, -66, -88, -82,-50,}{53, 85, 85, 53, 0, -53, -85, -85, -53, 0, 53, 85, 85, 53, 0, -53, -85,-85, -53, 0, 53, 85, 85, 53, 0, -53, -85, -85, -53, 0, 53, 85, 85, 53,0, -53, -85, -85, -53, 0, 53, 85, 85, 53, 0, -53, -85, -85, -53, 0, 53,85, 85, 53, 0, -53, -85, -85, -53, 0, 53, 85, 85, 53,}{56, 88, 80, 38, -21, -72, -90, -68, -17, 42, 82, 86, 53, -4, -60, -88,-78, -34, 26, 74, 90, 66, 13, -45, -84, -85, -50, 9, 63, 89, 77, 30,-30, -77, -89, -63, -9, 50, 85, 84, 45, -13, -66, -90, -74, -26, 34, 78,88, 60, 4, -53, -86, -82, -42, 17, 68, 90, 72, 21, -38, -80, -88, -56,}{60, 89, 74, 21, -42, -84, -84, -42, 21, 74, 89, 60, 0, -60, -89, -74,-21, 42, 84, 84, 42, -21, -74, -89, -60, 0, 60, 89, 74, 21, -42, -84,-84, -42, 21, 74, 89, 60, 0, -60, -89, -74, -21, 42, 84, 84, 42, -21,-74, -89, -60, 0, 60, 89, 74, 21, -42, -84, -84, -42, 21, 74, 89, 60,}{63, 90, 66, 4, -60, -90, -68, -9, 56, 89, 72, 13, -53, -88, -74, -17,50, 88, 77, 21, -45, -86, -78, -26, 42, 85, 80, 30, -38, -84, -82, -34,34, 82, 84, 38, -30, -80, -85, -42, 26, 78, 86, 45, -21, -77, -88, -50,17, 74, 88, 53, -13, -72, -89, -56, 9, 68, 90, 60, -4, -66, -90, -63,}{66, 90, 56, -13, -74, -88, -45, 26, 80, 84, 34, -38, -85, -78, -21, 50,88, 72, 9, -60, -90, -63, 4, 68, 89, 53, -17, -77, -86, -42, 30, 82, 82,30, -42, -86, -77, -17, 53, 89, 68, 4, -63, -90, -60, 9, 72, 88, 50,-21, -78, -85, -38, 34, 84, 80, 26, -45, -88, -74, -13, 56, 90, 66,}{68, 88, 45, -30, -84, -78, -17, 56, 90, 60, -13, -77, -85, -34, 42, 88,72, 4, -66, -89, -50, 26, 82, 80, 21, -53, -90, -63, 9, 74, 86, 38, -38,-86, -74, -9, 63, 90, 53, -21, -80, -82, -26, 50, 89, 66, -4, -72, -88,-42, 34, 85, 77, 13, -60, -90, -56, 17, 78, 84, 30, -45, -88, -68,}{72, 86, 34, -45, -89, -63, 13, 78, 82, 21, -56, -90, -53, 26, 84, 77,9, -66, -88, -42, 38, 88, 68, -4, -74, -85, -30, 50, 90, 60, -17, -80,-80, -17, 60, 90, 50, -30, -85, -74, -4, 68, 88, 38, -42, -88, -66, 9,77, 84, 26, -53, -90, -56, 21, 82, 78, 13, -63, -89, -45, 34, 86, 72,}{74, 84, 21, -60, -89, -42, 42, 89, 60, -21, -84, -74, 0, 74, 84, 21,-60, -89, -42, 42, 89, 60, -21, -84, -74, 0, 74, 84, 21, -60, -89, -42,42, 89, 60, -21, -84, -74, 0, 74, 84, 21, -60, -89, -42, 42, 89, 60,-21, -84, -74, 0, 74, 84, 21, -60, -89, -42, 42, 89, 60, -21, -84, -74,}{77, 80, 9, -72, -84, -17, 66, 86, 26, -60, -88, -34, 53, 90, 42, -45,-90, -50, 38, 89, 56, -30, -88, -63, 21, 85, 68, -13, -82, -74, 4, 78,78, 4, -74, -82, -13, 68, 85, 21, -63, -88, -30, 56, 89, 38, -50, -90,-45, 42, 90, 53, -34, -88, -60, 26, 86, 66, -17, -84, -72, 9, 80, 77,}{78, 77, -4, -80, -74, 9, 82, 72, -13, -84, -68, 17, 85, 66, -21, -86,-63, 26, 88, 60, -30, -88, -56, 34, 89, 53, -38, -90, -50, 42, 90, 45,-45, -90, -42, 50, 90, 38, -53, -89, -34, 56, 88, 30, -60, -88, -26, 63,86, 21, -66, -85, -17, 68, 84, 13, -72, -82, -9, 74, 80, 4, -77, -78,}{80, 72, -17, -86, -60, 34, 90, 45, -50, -89, -30, 63, 85, 13, -74, -78,4, 82, 68, -21, -88, -56, 38, 90, 42, -53, -88, -26, 66, 84, 9, -77,-77, 9, 84, 66, -26, -88, -53, 42, 90, 38, -56, -88, -21, 68, 82, 4,-78, -74, 13, 85, 63, -30, -89, -50, 45, 90, 34, -60, -86, -17, 72, 80,}{82, 66, -30, -90, -42, 56, 86, 13, -77, -74, 17, 88, 53, -45, -89, -26,68, 80, -4, -84, -63, 34, 90, 38, -60, -85, -9, 78, 72, -21, -88, -50,50, 88, 21, -72, -78, 9, 85, 60, -38, -90, -34, 63, 84, 4, -80, -68, 26,89, 45, -53, -88, -17, 74, 77, -13, -86, -56, 42, 90, 30, -66, -82,}{84, 60, -42, -89, -21, 74, 74, -21, -89, -42, 60, 84, 0, -84, -60, 42,89, 21, -74, -74, 21, 89, 42, -60, -84, 0, 84, 60, -42, -89, -21, 74,74, -21, -89, -42, 60, 84, 0, -84, -60, 42, 89, 21, -74, -74, 21, 89,42, -60, -84, 0, 84, 60, -42, -89, -21, 74, 74, -21, -89, -42, 60, 84,}{85, 53, -53, -85, 0, 85, 53, -53, -85, 0, 85, 53, -53, -85, 0, 85, 53,-53, -85, 0, 85, 53, -53, -85, 0, 85, 53, -53, -85, 0, 85, 53, -53, -85,0, 85, 53, -53, -85, 0, 85, 53, -53, -85, 0, 85, 53, -53, -85, 0, 85,53, -53, -85, 0, 85, 53, -53, -85, 0, 85, 53, -53, -85,}{86, 45, -63, -78, 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-4, -9, 21, -34, 45, -56,66, -74, 80, -85, 88, -90, 89, -86, 82, -77, 68, -60, 50, -38, 26, -13,}{9, -17, 26, -34, 42, -50, 56, -63, 68, -74, 78, -82, 85, -88, 89, -90,90, -88, 86, -84, 80, -77, 72, -66, 60, -53, 45, -38, 30, -21, 13, -4,-4, 13, -21, 30, -38, 45, -53, 60, -66, 72, -77, 80, -84, 86, -88, 90,-90, 89, -88, 85, -82, 78, -74, 68, -63, 56, -50, 42, -34, 26, -17, 9,}{4, -9, 13, -17, 21, -26, 30, -34, 38, -42, 45, -50, 53, -56, 60, -63,66, -68, 72, -74, 77, -78, 80, -82, 84, -85, 86, -88, 88, -89, 90, -90,90, -90, 89, -88, 88, -86, 85, -84, 82, -80, 78, -77, 74, -72, 68, -66,63, -60, 56, -53, 50, -45, 42, -38, 34, -30, 26, -21, 17, -13, 9, -4,}.

In an embodiment, one or more 8-bit DCT-5 transform cores may be usedfor the primary transform. As an example, the 8-bit DCT-5 transformcores may be:

4-Point DCT-5:

{47, 69, 69, 69,}

{69, 61, -21, -87,}

{69, -21, -87, 61,}

{69, -87, 61, -21,}

8-Point DCT-5:

{47, 66, 66, 66, 66, 66, 66, 66,}

{66, 86, 62, 29, -10, -47, -76, -91,}

{66, 62, -10, -76, -91, -47, 29, 86,}

{66, 29, -76, -76, 29, 93, 29, -76,}

{66, -10, -91, 29, 86, -47, -76, 62,}

{66, -47, -47, 93, -47, -47, 93, -47,}

{66, -76, 29, 29, -76, 93, -76, 29,}

{66, -91, 86, -76, 62, -47, 29, -10,}

16-Point DCT-5:

{45, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65,}

{65, 89, 85, 75, 64, 48, 33, 15, -5, -23, -39, -56, -71, -81, -88, -91,}

{65, 85, 64, 33, -5, -39, -71, -88, -91, -81, -56, -23, 15, 48, 75, 89,}

{65, 75, 33, -23, -71, -91, -81, -39, 15, 64, 89, 85, 48, -5, -56, -88,}

{65, 64, -5, -71, -91, -56, 15, 75, 89, 48, -23, -81, -88, -39, 33, 85,}

{65, 48, -39, -91, -56, 33, 89, 64, -23, -88, -71, 15, 85, 75, -5, -81,}

{65, 33, -71, -81, 15, 89, 48, -56, -88, -5, 85, 64, -39, -91, -23, 75,}

{65, 15, -88, -39, 75, 64, -56, -81, 33, 89, -5, -91, -23, 85, 48, -71,}

{65, -5, -91, 15, 89, -23, -88, 33, 85, -39, -81, 48, 75, -56, -71, 64,}

{65, -23, -81, 64, 48, -88, -5, 89, -39, -71, 75, 33, -91, 15, 85, -56,}

{65, -39, -56, 89, -23, -71, 85, -5, -81, 75, 15, -88, 64, 33, -91, 48,}

{65, -56, -23, 85, -81, 15, 64, -91, 48, 33, -88, 75, -5, -71, 89, -39,}

{65, -71, 15, 48, -88, 85, -39, -23, 75, -91, 64, -5, -56, 89, -81, 33,}

{65, -81, 48, -5, -39, 75, -91, 85, -56, 15, 33, -71, 89, -88, 64, -23,}

{65, -88, 75, -56, 33, -5, -23, 48, -71, 85, -91, 89, -81, 64, -39, 15,}

{65, -91, 89, -88, 85, -81, 75, -71, 64, -56, 48, -39, 33, -23, 15, -5,}

32-Point DCT-5:

{46, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65,65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65,}

{65, 91, 89, 87, 84, 80, 75, 70, 64, 57, 49, 42, 33, 25, 16, 7, -2, -12,-20, -29, -38, -46, -53, -60, -67, -72, -78, -82, -86, -89, -90, -91,}

{65, 89, 84, 75, 64, 49, 33, 16, -2, -20, -38, -53, -67, -78, -86, -90,-91, -89, -82, -72, -60, -46, -29, -12, 7, 25, 42, 57, 70, 80, 87, 91,}

{65, 87, 75, 57, 33, 7, -20, -46, -67, -82, -90, -90, -82, -67, -46,-20, 7, 33, 57, 75, 87, 92, 87, 75, 57, 33, 7, -20, -46, -67, -82, -90,}

{65, 84, 64, 33, -2, -38, -67, -86, -91, -82, -60, -29, 7, 42, 70, 87,91, 80, 57, 25, -12, -46, -72, -89, -90, -78, -53, -20, 16, 49, 75, 89,}

{65, 80, 49, 7, -38, -72, -90, -86, -60, -20, 25, 64, 87, 89, 70, 33,-12, -53, -82, -91, -78, -46, -2, 42, 75, 91, 84, 57, 16, -29, -67,-89,}

{65, 75, 33, -20, -67, -90, -82, -46, 7, 57, 87, 87, 57, 7, -46, -82,-90, -67, -20, 33, 75, 92, 75, 33, -20, -67, -90, -82, -46, 7, 57, 87,}

{65, 70, 16, -46, -86, -86, -46, 16, 70, 92, 70, 16, -46, -86, -86, -46,16, 70, 92, 70, 16, -46, -86, -86, -46, 16, 70, 92, 70, 16, -46, -86,}

{65, 64, -2, -67, -91, -60, 7, 70, 91, 57, -12, -72, -90, -53, 16, 75,89, 49, -20, -78, -89, -46, 25, 80, 87, 42, -29, -82, -86, -38, 33, 84,}

{65, 57, -20, -82, -82, -20, 57, 92, 57, -20, -82, -82, -20, 57, 92, 57,-20, -82, -82, -20, 57, 92, 57, -20, -82, -82, -20, 57, 92, 57, -20,-82,}

{65, 49, -38, -90, -60, 25, 87, 70, -12, -82, -78, -2, 75, 84, 16, -67,-89, -29, 57, 91, 42, -46, -91, -53, 33, 89, 64, -20, -86, -72, 7, 80,}

{65, 42, -53, -90, -29, 64, 87, 16, -72, -82, -2, 80, 75, -12, -86, -67,25, 89, 57, -38, -91, -46, 49, 91, 33, -60, -89, -20, 70, 84, 7, -78,}

{65, 33, -67, -82, 7, 87, 57, -46, -90, -20, 75, 75, -20, -90, -46, 57,87, 7, -82, -67, 33, 92, 33, -67, -82, 7, 87, 57, -46, -90, -20, 75,}

{65, 25, -78, -67, 42, 89, 7, -86, -53, 57, 84, -12, -90, -38, 70, 75,-29, -91, -20, 80, 64, -46, -89, -2, 87, 49, -60, -82, 16, 91, 33, -72,}

{65, 16, -86, -46, 70, 70, -46, -86, 16, 92, 16, -86, -46, 70, 70, -46,-86, 16, 92, 16, -86, -46, 70, 70, -46, -86, 16, 92, 16, -86, -46, 70,}

{65, 7, -90, -20, 87, 33, -82, -46, 75, 57, -67, -67, 57, 75, -46, -82,33, 87, -20, -90, 7, 92, 7, -90, -20, 87, 33, -82, -46, 75, 57, -67,}

{65, -2, -91, 7, 91, -12, -90, 16, 89, -20, -89, 25, 87, -29, -86, 33,84, -38, -82, 42, 80, -46, -78, 49, 75, -53, -72, 57, 70, -60, -67, 64,}

{65, -12, -89, 33, 80, -53, -67, 70, 49, -82, -29, 89, 7, -91, 16, 87,-38, -78, 57, 64, -72, -46, 84, 25, -90, -2, 91, -20, -86, 42, 75, -60,}

{65, -20, -82, 57, 57, -82, -20, 92, -20, -82, 57, 57, -82, -20, 92,-20, -82, 57, 57, -82, -20, 92, -20, -82, 57, 57, -82, -20, 92, -20,-82, 57,}

{65, -29, -72, 75, 25, -91, 33, 70, -78, -20, 91, -38, -67, 80, 16, -90,42, 64, -82, -12, 89, -46, -60, 84, 7, -89, 49, 57, -86, -2, 87, -53,}

{65, -38, -60, 87, -12, -78, 75, 16, -89, 57, 42, -91, 33, 64, -86, 7,80, -72, -20, 89, -53, -46, 91, -29, -67, 84, -2, -82, 70, 25, -90, 49,}

{65, -46, -46, 92, -46, -46, 92, -46, -46, 92, -46, -46, 92, -46, -46,92, -46, -46, 92, -46, -46, 92, -46, -46, 92, -46, -46, 92, -46, -46,92, -46,}

{65, -53, -29, 87, -72, -2, 75, -86, 25, 57, -91, 49, 33, -89, 70, 7,-78, 84, -20, -60, 91, -46, -38, 89, -67, -12, 80, -82, 16, 64, -90,42,}

{65, -60, -12, 75, -89, 42, 33, -86, 80, -20, -53, 91, -67, -2, 70, -90,49, 25, -82, 84, -29, -46, 89, -72, 7, 64, -91, 57, 16, -78, 87, -38,}

{65, -67, 7, 57, -90, 75, -20, -46, 87, -82, 33, 33, -82, 87, -46, -20,75, -90, 57, 7, -67, 92, -67, 7, 57, -90, 75, -20, -46, 87, -82, 33,}

{65, -72, 25, 33, -78, 91, -67, 16, 42, -82, 89, -60, 7, 49, -86, 87,-53, -2, 57, -89, 84, -46, -12, 64, -90, 80, -38, -20, 70, -91, 75,-29,}

{65, -78, 42, 7, -53, 84, -90, 70, -29, -20, 64, -89, 87, -60, 16, 33,-72, 91, -82, 49, -2, -46, 80, -91, 75, -38, -12, 57, -86, 89, -67, 25,}

{65, -82, 57, -20, -20, 57, -82, 92, -82, 57, -20, -20, 57, -82, 92,-82, 57, -20, -20, 57, -82, 92, -82, 57, -20, -20, 57, -82, 92, -82, 57,-20,}

{65, -86, 70, -46, 16, 16, -46, 70, -86, 92, -86, 70, -46, 16, 16, -46,70, -86, 92, -86, 70, -46, 16, 16, -46, 70, -86, 92, -86, 70, -46, 16,}

{65, -89, 80, -67, 49, -29, 7, 16, -38, 57, -72, 84, -90, 91, -86, 75,-60, 42, -20, -2, 25, -46, 64, -78, 87, -91, 89, -82, 70, -53, 33, -12,}

{65, -90, 87, -82, 75, -67, 57, -46, 33, -20, 7, 7, -20, 33, -46, 57,-67, 75, -82, 87, -90, 92, -90, 87, -82, 75, -67, 57, -46, 33, -20, 7,}

{65, -91, 91, -90, 89, -89, 87, -86, 84, -82, 80, -78, 75, -72, 70, -67,64, -60, 57, -53, 49, -46, 42, -38, 33, -29, 25, -20, 16, -12, 7, -2,}

64-Point DCT-5:

{44, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64,64, 64, 64, 64, 64, 64, 64, 64, 64, 64,}{64, 91, 90, 90, 90, 88, 87, 85, 84, 82, 80, 78, 75, 72, 70, 66, 64, 61,58, 54, 50, 46, 42, 39, 34, 29, 25, 22, 17, 13, 7, 4, -1, -6, -10, -15,-19, -24, -28, -32, -36, -40, -44, -48, -51, -55, -59, -63, -65, -68,-70, -73, -77, -79, -81, -83, -85, -86, -88, -89, -90, -90, -91, -91,}{64, 90, 90, 87, 84, 80, 75, 70, 64, 58, 50, 42, 34, 25, 17, 7, -1, -10,-19, -28, -36, -44, -51, -59, -65, -70, -77, -81, -85, -88, -90, -91,-91, -90, -89, -86, -83, -79, -73, -68, -63, -55, -48, -40, -32, -24, -15, -6, 4, 13, 22, 29, 39, 46, 54, 61, 66, 72, 78, 82, 85, 88, 90, 91,}{64, 90, 87, 82, 75, 66, 58, 46, 34, 22, 7, -6, -19, -32, -44, -55, -65,-73, -81, -86, -90, -91, -90, -88, -83, -77, -68, -59, -48, -36, -24,-10, 4, 17, 29, 42, 54, 64, 72, 80, 85, 90, 91, 90, 88, 84, 78, 70, 61,50, 39, 25, 13, 4, 45, -28, -40, -51, -63, -70, -79, -85, -89, -91,}{64, 90, 84, 75, 64, 50, 34, 17, -1, -19, -36, -51, -65, -77, -85, -90,-91, -89, -83, -73, -63, -48, -32, -15, 4, 22, 39, 54, 66, 78, 85, 90,91, 88, 82, 72, 61, 46, 29, 13, -6, -24, -40, -55, -68, -79, -86, -90,-91, -88, -81, -70, -59, -44, -28, -10, 7, 25, 42, 58, 70, 80, 87, 90,}{64, 88, 80, 66, 50, 29, 7, -15, -36, -55, -70, -83, -90, -91, -86, -77,-63, -44, -24, -1, 22, 42, 61, 75, 85, 90, 90, 84, 72, 58, 39, 17, -6,-28, -48, -65, -79, -88, -91, -89, -81, -68, -51, -32, -10, 13, 34, 54,70, 82, 90, 91, 87, 78, 64, 46, 25, 4, -19, -40, -59, -73, -85, -90,}{64, 87, 75, 58, 34, 7, -19, -44, -65, -81, -90, -90, -83, -68, -48,-24, 4, 29, 54, 72, 85, 91, 88, 78, 61, 39, 13, -15, -40, -63, -79, -89,-91, -85, -70, -51, -28, -1, 25, 50, 70, 84, 90, 90, 80, 64, 42, 17,-10, -36, -59, -77, -88, -91, -86, -73, -55, -32, -6, 22, 46, 66, 82,90,}{64, 85, 70, 46, 17, -15, -44, -68, -85, -91, -86, -70, -48, -19, 13,42, 66, 84, 91, 87, 72, 50, 22, -10, -40, -65, -83, -91, -88, -73, -51,-24, 7, 39, 64, 82, 90, 88, 75, 54, 25, -6, -36, -63, -81, -90, -89,-77, -55, -28, 4, 34, 61, 80, 90, 90, 78, 58, 29, -1, -32, -59, -79,-90,}{64, 84, 64, 34, -1, -36, -65, -85, -91, -83, -63, -32, 4, 39, 66, 85,91, 82, 61, 29, -6, -40, -68, -86, -91, -81, -59, -28, 7, 42, 70, 87,90, 80, 58, 25, -10, -44, -70, -88, -90, -79, -55, -24, 13, 46, 72, 88,90, 78, 54, 22, -15, -48, -73, -89, -90, -77, -51, -19, 17, 50, 75, 90,}{64, 82, 58, 22, -19, -55, -81, -91, -83, -59, -24, 17, 54, 80, 91, 84,61, 25, -15, -51, -79, -91, -85, -63, -28, 13, 50, 78, 90, 85, 64, 29,-10, -48, -77, -90, -86, -65, -32, 7, 46, 75, 90, 87, 66, 34, -6, -44,-73, -90, -88, -68, -36, 4, 42, 72, 90, 88, 70, 39, -1, -40, -70, -89,}{64, 80, 50, 7, -36, -70, -90, -86, -63, -24, 22, 61, 85, 90, 72, 39,-6, -48, -79, -91, -81, -51, -10, 34, 70, 90, 87, 64, 25, -19, -59, -85,-90, -73, -40, 4, 46, 78, 91, 82, 54, 13, -32, -68, -89, -88, -65, -28,17, 58, 84, 90, 75, 42, -1, -44, -77, -91, -83, -55, -15, 29, 66, 88,}{64, 78, 42, -6, -51, -83, -90, -70, -32, 17, 61, 87, 88, 64, 22, -28,-68, -90, -85, -55, -10, 39, 75, 91, 80, 46, -1, -48, -81, -91, -73,-36, 13, 58, 85, 90, 66, 25, -24, -65, -89, -86, -59, -15, 34, 72, 90,82, 50, 4, -44, -79, -91, -77, -40, 7, 54, 84, 90, 70, 29, -19, -63,-88,}{64, 75, 34, -19, -65, -90, -83, -48, 4, 54, 85, 88, 61, 13, -40, -79,-91, -70, -28, 25, 70, 90, 80, 42, -10, -59, -88, -86, -55, -6, 46, 82,90, 66, 22, -32, -73, -91, -77, -36, 17, 64, 90, 84, 50, -1, -51, -85,-89, -63, -15, 39, 78, 91, 72, 29, -24, -68, -90, -81, -44, 7, 58, 87,}{64, 72, 25, -32, -77, -91, -68, -19, 39, 80, 90, 64, 13, -44, -83, -89,-59, -6, 50, 85, 87, 54, -1, -55, -88, -85, -48, 7, 61, 90, 82, 42, -15,-65, -90, -79, -36, 22, 70, 91, 75, 29, -28, -73, -91, -70, -24, 34, 78,90, 66, 17, -40, -81, -90, -63, -10, 46, 84, 88, 58, 4, -51, -86,}{64, 70, 17, -44, -85, -86, -48, 13, 66, 91, 72, 22, -40, -83, -88, -51,7, 64, 90, 75, 25, -36, -81, -89, -55, 4, 61, 90, 78, 29, -32, -79, -90,-59, -1, 58, 90, 80, 34, -28, -77, -90, -63, -6, 54, 88, 82, 39, -24,-73, -91, -65, -10, 50, 87, 84, 42, -19, -70, -91, -68, -15, 46, 85,}{64, 66, 7, -55, -90, -77, -24, 42, 85, 84, 39, -28, -79, -89, -51, 13,70, 91, 64, 4, -59, -90, -73, -19, 46, 87, 82, 34, -32, -81, -88, -48,17, 72, 90, 61, -1, -63, -91, -70, -15, 50, 88, 80, 29, -36, -83, -86,-44, 22, 75, 90, 58, -6, -65, -91, -68, -10, 54, 90, 78, 25, -40, -85,}{64, 64, -1, -65, -91, -63, 4, 66, 91, 61, -6, -68, -91, -59, 7, 70, 90,58, -10, -70, -90, -55, 13, 72, 90, 54, -15, -73, -90, -51, 17, 75, 90,50, -19, -77, -89, -48, 22, 78, 88, 46, -24, -79, -88, -44, 25, 80, 87,42, -28, -81, -86, -40, 29, 82, 85, 39, -32, -83, -85, -36, 34, 84,}{64, 61, -10, -73, -89, -44, 29, 84, 82, 25, -48, -90, -70, -6, 64, 91,58, -15, -77, -88, -40, 34, 85, 80, 22, -51, -90, -68, -1, 66, 90, 54,-19, -79, -86, -36, 39, 87, 78, 17, -55, -91, -65, 4, 70, 90, 50, -24,-81, -85, -32, 42, 88, 75, 13, -59, -91, -63, 7, 72, 90, 46, -28, -83,}{64, 58, -19, -81, -83, -24, 54, 91, 61, -15, -79, -85, -28, 50, 90, 64,-10, -77, -86, -32, 46, 90, 66, -6, -73, -88, -36, 42, 90, 70, -1, -70,-89, -40, 39, 88, 72, 4, -68, -90, -44, 34, 87, 75, 7, -65, -90, -48,29, 85, 78, 13, -63, -91, -51, 25, 84, 80, 17, -59, -91, -55, 22, 82,}{64, 54, -28, -86, -73, -1, 72, 87, 29, -51, -91, -55, 25, 85, 75, 4,-70, -88, -32, 50, 91, 58, -24, -85, -77, -6, 70, 88, 34, -48, -91, -59,22, 84, 78, 7, -68, -89, -36, 46, 90, 61, -19, -83, -79, -10, 66, 90,39, -44, -90, -63, 17, 82, 80, 13, -65, -90, -40, 42, 90, 64, -15, -81,}{64, 50, -36, -90, -63, 22, 85, 72, -6, -79, -81, -10, 70, 87, 25, -59,-90, -40, 46, 91, 54, -32, -89, -65, 17, 84, 75, -1, -77, -83, -15, 66,88, 29, -55, -91, -44, 42, 90, 58, -28, -88, -68, 13, 82, 78, 4, -73,-85, -19, 64, 90, 34, -51, -91, -48, 39, 90, 61, -24, -86, -70, 7, 80,}{64, 46, -44, -91, -48, 42, 91, 50, -40, -91, -51, 39, 90, 54, -36, -90,-55, 34, 90, 58, -32, -90, -59, 29, 90, 61, -28, -89, -63, 25, 88, 64,-24, -88, -65, 22, 87, 66, -19, -86, -68, 17, 85, 70, -15, -85, -70, 13,84, 72, -10, -83, -73, 7, 82, 75, -6, -81, -77, 4, 80, 78, -1, -79,}{64, 42, -51, -90, -32, 61, 88, 22, -68, -85, -10, 75, 80, -1, -81, -73,13, 85, 66, -24, -89, -59, 34, 90, 50, -44, -91, -40, 54, 90, 29, -63,-88, -19, 70, 84, 7, -77, -79, 4, 82, 72, -15, -86, -65, 25, 90, 58,-36, -91, -48, 46, 91, 39, -55, -90, -28, 64, 87, 17, -70, -83, -6, 78,}{64, 39, -59, -88, -15, 75, 78, -10, -86, -63, 34, 91, 42, -55, -89,-19, 72, 80, -6, -85, -65, 29, 90, 46, -51, -90, -24, 70, 82, -1, -83,-68, 25, 90, 50, -48, -90, -28, 66, 84, 4, -81, -70, 22, 90, 54, -44,-91, -32, 64, 85, 7, -79, -73, 17, 88, 58, -40, -91, -36, 61, 87, 13,-77,}{64, 34, -65, -83, 4, 85, 61, -40, -91, -28, 70, 80, -10, -88, -55, 46,90, 22, -73, -77, 17, 90, 50, -51, -89, -15, 78, 72, -24, -90, -44, 58,87, 7, -81, -68, 29, 91, 39, -63, -85, -1, 84, 64, -36, -91, -32, 66,82, -6, -86, -59, 42, 90, 25, -70, -79, 13, 88, 54, -48, -90, -19, 75,}{64, 29, -70, -77, 22, 90, 39, -65, -81, 13, 90, 46, -59, -85, 4, 87,54, -51, -88, -6, 84, 61, -44, -90, -15, 80, 66, -36, -91, -24, 75, 72,-28, -91, -32, 70, 78, -19, -90, -40, 64, 82, -10, -89, -48, 58, 85, -1,-86, -55, 50, 88, 7, -83, -63, 42, 90, 17, -79, -68, 34, 91, 25, -73,}{64, 25, -77, -68, 39, 90, 13, -83, -59, 50, 87, -1, -88, -48, 61, 82,-15, -90, -36, 70, 75, -28, -91, -24, 78, 66, -40, -90, -10, 84, 58,-51, -86, 4, 88, 46, -63, -81, 17, 90, 34, -70, -73, 29, 91, 22, -79,-65, 42, 90, 7, -85, -55, 54, 85, -6, -89, -44, 64, 80, -19, -91, -32,72,}{64, 22, -81, -59, 54, 84, -15, -91, -28, 78, 64, -48, -86, 7, 90, 34,-73, -68, 42, 88, -1, -89, -40, 70, 72, -36, -90, -6, 87, 46, -65, -77,29, 90, 13, -85, -51, 61, 80, -24, -91, -19, 82, 58, -55, -83, 17, 91,25, -79, -63, 50, 85, -10, -90, -32, 75, 66, -44, -88, 4, 90, 39, -70,}{64, 17, -85, -48, 66, 72, -40, -88, 7, 90, 25, -81, -55, 61, 78, -32,-90, -1, 90, 34, -77, -63, 54, 82, -24, -91, -10, 87, 42, -70, -68, 46,85, -15, -91, -19, 84, 50, -65, -73, 39, 88, -6, -90, -28, 80, 58, -59,-79, 29, 90, 4, -89, -36, 75, 64, -51, -83, 22, 91, 13, -86, -44, 70,}{64, 13, -88, -36, 78, 58, -63, -73, 42, 85, -19, -91, -6, 90, 29, -81,-51, 66, 70, -48, -83, 25, 90, -1, -90, -24, 84, 46, -70, -65, 54, 80,-32, -89, 7, 91, 17, -86, -40, 75, 61, -59, -77, 39, 87, -15, -91, -10,88, 34, -79, -55, 64, 72, -44, -85, 22, 90, 4, -90, -28, 82, 50, -68,}{64, 7, -90, -24, 85, 39, -79, -51, 70, 64, -59, -73, 46, 82, -32, -88,17, 90, -1, -91, -15, 88, 29, -83, -44, 75, 58, -65, -68, 54, 78, -40,-85, 25, 90, -10, -91, -6, 90, 22, -86, -36, 80, 50, -70, -63, 61, 72,-48, -81, 34, 87, -19, -90, 4, 91, 13, -89, -28, 84, 42, -77, -55, 66,}{64, 4, -91, -10, 90, 17, -89, -24, 87, 29, -85, -36, 82, 42, -79, -48,75, 54, -70, -59, 66, 64, -63, -68, 58, 72, -51, -77, 46, 80, -40, -83,34, 85, -28, -88, 22, 90, -15, -90, 7, 91, -1, -91, -6, 90, 13, -90,-19, 88, 25, -86, -32, 84, 39, -81, -44, 78, 50, -73, -55, 70, 61, -65,}{64, 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The techniques for encoding/decoding can be implemented by one or moreprocessors that may execute computer software having computer-readableinstructions that may be physically stored in one or morecomputer-readable media (e.g., a hard disk drive). For example, FIG. 7shows a computer system 700 suitable for implementing certainembodiments of the disclosed subject matter.

The computer software can be coded using any suitable machine code orcomputer language, that may be subject to assembly, compilation,linking, or like mechanisms to create code comprising instructions thatcan be executed directly, or through interpretation, micro-codeexecution, and the like, by computer central processing units (CPUs),Graphics Processing Units (GPUs), and the like.

The instructions can be executed on various types of computers orcomponents thereof, including, for example, personal computers, tabletcomputers, servers, smartphones, gaming devices, internet of thingsdevices, and the like.

The components shown in FIG. 7 for computer system 700 are exemplary innature and are not intended to suggest any limitation as to the scope ofuse or functionality of the computer software implementing embodimentsof the present disclosure. Neither should the configuration ofcomponents be interpreted as having any dependency or requirementrelating to any one or combination of components illustrated in theexemplary embodiment of a computer system 700.

Computer system 700 may include certain human interface input devices.Such a human interface input device may be responsive to input by one ormore human users through, for example, tactile input (such as:keystrokes, swipes, data glove movements), audio input (such as: voice,clapping), visual input (such as: gestures), olfactory input (notdepicted). The human interface devices can also be used to capturecertain media not necessarily directly related to conscious input by ahuman, such as audio (such as: speech, music, ambient sound), images(such as: scanned images, photographic images obtain from a still imagecamera), video (such as two-dimensional video, three-dimensional videoincluding stereoscopic video).

Input human interface devices may include one or more of (only one ofeach depicted): keyboard 701, mouse 702, trackpad 703, touch screen 710,data-glove 704, joystick 705, microphone 706, scanner 707, camera 708.

Computer system 700 may also include certain human interface outputdevices. Such human interface output devices may be stimulating thesenses of one or more human users through, for example, tactile output,sound, light, and smell/taste. Such human interface output devices mayinclude tactile output devices (for example tactile feedback by thetouch-screen 710, data-glove 704, or joystick 705, but there can also betactile feedback devices that do not serve as input devices), audiooutput devices (such as: speakers 709, headphones (not depicted)),visual output devices (such as screens 710 to include CRT screens, LCDscreens, plasma screens, OLED screens, each with or without touch-screeninput capability, each with or without tactile feedback capability—someof which may be capable to output two dimensional visual output or morethan three dimensional output through means such as stereographicoutput; virtual-reality glasses (not depicted), holographic displays andsmoke tanks (not depicted)), and printers (not depicted).

Computer system 700 can also include human accessible storage devicesand their associated media such as optical media including CD/DVD ROM/RW720 with CD/DVD or the like media 721, thumb-drive 722, removable harddrive or solid state drive 723, legacy magnetic media such as tape andfloppy disc (not depicted), specialized ROM/ASIC/PLD based devices suchas security dongles (not depicted), and the like.

Those skilled in the art should also understand that term “computerreadable media” or “computer readable medium” as used in connection withthe presently disclosed subject matter does not encompass transmissionmedia, carrier waves, or other transitory signals.

Computer system 700 can also include interface to one or morecommunication networks. Networks can for example be wireless, wireline,optical. Networks can further be local, wide-area, metropolitan,vehicular and industrial, real-time, delay-tolerant, and so on. Examplesof networks include local area networks such as Ethernet, wireless LANs,cellular networks to include GSM, 3G, 4G, 5G, LTE and the like, TVwireline or wireless wide area digital networks to include cable TV,satellite TV, and terrestrial broadcast TV, vehicular and industrial toinclude CANBus, and so forth. Certain networks commonly require externalnetwork interface adapters that attached to certain general purpose dataports or peripheral buses (749) (such as, for example USB ports of thecomputer system 700; others are commonly integrated into the core of thecomputer system 700 by attachment to a system bus as described below(for example Ethernet interface into a PC computer system or cellularnetwork interface into a smartphone computer system). Using any of thesenetworks, computer system 700 can communicate with other entities. Suchcommunication can be uni-directional, receive only (for example,broadcast TV), uni-directional send-only (for example CANbus to certainCANbus devices), or bi-directional, for example to other computersystems using local or wide area digital networks. Certain protocols andprotocol stacks can be used on each of those networks and networkinterfaces as described above.

Aforementioned human interface devices, human-accessible storagedevices, and network interfaces can be attached to a core 740 of thecomputer system 700.

The core 740 can include one or more Central Processing Units (CPU) 741,Graphics Processing Units (GPU) 742, specialized programmable processingunits in the form of Field Programmable Gate Areas (FPGA) 743, hardwareaccelerators for certain tasks 744, and so forth. These devices, alongwith Read-only memory (ROM) 745, Random-access memory 746, internal massstorage such as internal non-user accessible hard drives, SSDs, and thelike 747, may be connected through a system bus 748. In some computersystems, the system bus 748 can be accessible in the form of one or morephysical plugs to enable extensions by additional CPUs, GPU, and thelike. The peripheral devices can be attached either directly to thecore's system bus 748, or through a peripheral bus 749. Architecturesfor a peripheral bus include PCI, USB, and the like.

CPUs 741, GPUs 742, FPGAs 743, and accelerators 744 can execute certaininstructions that, in combination, can make up the aforementionedcomputer code. That computer code can be stored in ROM 745 or RAM 746.Transitional data can be also be stored in RAM 746, whereas permanentdata can be stored for example, in the internal mass storage 747. Faststorage and retrieve to any of the memory devices can be enabled throughthe use of cache memory, that can be closely associated with one or moreCPU 741, GPU 742, mass storage 747, ROM 745, RAM 746, and the like.

The computer readable media can have computer code thereon forperforming various computer-implemented operations. The media andcomputer code can be those specially designed and constructed for thepurposes of the present disclosure, or they can be of the kind wellknown and available to those having skill in the computer software arts.

As an example and not by way of limitation, the computer system havingarchitecture 700, and specifically the core 740 can providefunctionality as a result of processor(s) (including CPUs, GPUs, FPGA,accelerators, and the like) executing software embodied in one or moretangible, computer-readable media. Such computer-readable media can bemedia associated with user-accessible mass storage as introduced above,as well as certain storage of the core 740 that are of non-transitorynature, such as core-internal mass storage 747 or ROM 745. The softwareimplementing various embodiments of the present disclosure can be storedin such devices and executed by core 740. A computer-readable medium caninclude one or more memory devices or chips, according to particularneeds. The software can cause the core 740 and specifically theprocessors therein (including CPU, GPU, FPGA, and the like) to executeparticular processes or particular parts of particular processesdescribed herein, including defining data structures stored in RAM 746and modifying such data structures according to the processes defined bythe software. In addition or as an alternative, the computer system canprovide functionality as a result of logic hardwired or otherwiseembodied in a circuit (for example: accelerator 744), which can operatein place of or together with software to execute particular processes orparticular parts of particular processes described herein. Reference tosoftware can encompass logic, and vice versa, where appropriate.Reference to a computer-readable media can encompass a circuit (such asan integrated circuit (IC)) storing software for execution, a circuitembodying logic for execution, or both, where appropriate. The presentdisclosure encompasses any suitable combination of hardware andsoftware.

While this disclosure has described several exemplary embodiments, thereare alterations, permutations, and various substitute equivalents, whichfall within the scope of the disclosure. It will thus be appreciatedthat those skilled in the art will be able to devise numerous systemsand methods which, although not explicitly shown or described herein,embody the principles of the disclosure and are thus within the spiritand scope thereof.

The invention claimed is:
 1. A method of decoding or encoding, themethod comprising: receiving information regarding a target file forencoding or decoding, the target file for encoding or decoding being oneof: a compressed video or image data file or an uncompressed video orimage data file; determining, for the encoding or decoding of the targetfile, whether to use a first transform core matrix that is of a firstsize type or a second transform core matrix that is of a second sizetype that is smaller than the first size type, wherein the first sizetype is the 8-bit primary transform core matrix, which has the corematrix that is represented by 8-bits and that has a size that is64-point or larger; when a result of the determination is to use thefirst transform core matrix of the first size type, causing ortransmitting information that causes the target file for encoding ordecoding to be encoded or decoded using the 8-bit primary transform corematrix that has the first size type that is 64-point or larger; and whena result of the determination is to use the second transform core matrixof the second size type that is smaller than the first size type:extracting the second transform core matrix of the second size type fromthe 8-bit primary transform core matrix that is 64-point or larger andcausing or transmitting information that causes the target file forencoding or decoding to be encoded or decoded using the extracted secondtransform core matrix of the second type, wherein when the firsttransform core matrix is a 64-point, 8-bit primary transform corematrix, which is constructed using thirty-three integers which are usedto construct a 32-point, 8-bit primary transform core matrix and anadditional thirty-two integers that include: bf, bg, bh, bi, bj, bk, bl,bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb, cc, cd,ce, cf, cg, ch, ci, cj and ck plus their respective sign changes, andthe additional thirty-two integers {bf, bg, bh, bi, bj, bk, bl, bm, bn,bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb, cc, cd, ce, cf,cg, ch, ci, cj, ck} are equal to {91, 90, 90, 90, 88, 87, 86, 84, 83,81, 79, 77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44, 41, 37, 33, 28, 24,20, 15, 11, 7, 2}.
 2. The method of claim 1, wherein the first primarytransform core matrix is a Versatile Video Coding (VVC) standardtransform.
 3. The method of claim 1, wherein the second primarytransform core matrix is one of DCT-8, DST-1 or DCT-5.
 4. The methodaccording to claim 1, wherein when deriving the N-point 8-bit DCT-8primary transform core matrix or the N-point 8-bit DST-7 transform corematrix, the set of M unique numbers which are used to construct theN-point DCT-8 core matrix is the same set of M unique numbers whichconstruct the N-point DST-7 core matrix.
 5. The method according toclaim 4, wherein M equals to N.
 6. The method according to claim 1,wherein a left half or a right half of even or odd rows of the 8-bitprimary transform core matrix form a matrix which is similar to thesecond transform core matrix.
 7. The method according to claim 6,wherein the 8-bit primary transform core matrix is one of: DCT-2, DCT-5,DCT-8, DST-1 and DST-7, and the second transform core matrix is one of:DCT-2, DCT-5, DCT-8, DST-1 and DST-7.
 8. The method according to claim7, wherein the 8-bit primary transform core matrix and the secondtransform core matrix are the same one of: DCT-2, DCT-5, DCT-8, DST-1and DST-7.
 9. The method according to claim 8, wherein the 8-bit primarytransform core matrix is an 8-bit, 64-point DCT-2 transform matrix andthe second transform core matrix is an 8-bit, 32-point DCT-2 transformmatrix.
 10. The method according to claim 1, wherein the 8-bit, N-pointprimary transform core matrix, which is composed by M unique numbers, isderived by the following steps: deriving an initial transform corematrix T1 by scaling an original transform core which is composed byfloating-point numbers T0 using a scaling factor 2B−2×log 2(N) such thatT1=T0×2^(B-2)×log₂(N), where B=8, and the row vectors of T0, which arebasis vectors, are orthogonal to each other, and the norm of each rowvector is 1; deriving an adjusted transform core matrix T2 by addingoffsets on a selected set of the M unique numbers which are used toconstruct the adjusted primary transform core matrix T2; calculating acost value C as the sum of absolute values of all elements of(T1*T1^(T)−2^(2B-4)×log₂(N)²×I), where I is an N×N identity matrix; andupdating the adjusted transform matrix T2 by adding the calculatedoffset values to T2, which minimizes the cost value to T2, which isoutput as the 8-bit, N-point primary transform core matrix.
 11. Themethod according to claim 10, further comprising: after minimizing thecost value C by trying all possible combinations of offset values on aselected set of M unique numbers, T2 is output as the 8-bit, N-pointprimary transform core matrix.
 12. An apparatus for utilizing an 8-bitprimary transform core matrix to decode compressed video or image dataor encode uncompressed video or image data, the apparatus comprising: atleast one memory configured to store computer program code; and at leastone processor configured to access the at least one memory and operateaccording to the computer program code, the computer program codecomprising: first applying code configured to cause the at least oneprocessor to receive information regarding a target file for encoding ordecoding, the target file for encoding or decoding being one of: acompressed video or image data file or an uncompressed video or imagedata file; second applying code configured to cause the at least oneprocessor to determine, for the encoding or decoding of the target file,whether to use a first transform core matrix that is of a first sizetype or a second transform core matrix that is of a second size typethat is smaller than the first size type, wherein the first size type isthe 8-bit primary transform core matrix, which has the core matrix thatis represented by 8-bits and that has a size that is 64-point or larger;third applying code configured to cause the at least one processor towhen a result of the determination is to use the first transform corematrix of the first size type, cause or transmit information that causesthe target file for encoding or decoding to be encoded or decoded usingthe 8-bit primary transform core matrix that has the size that is64-point or larger; and fourth applying code configured to cause the atleast one processor to when a result of the determination is to use thesecond transform core matrix of the second size type that is smallerthan the first size type: extract the second transform core matrix ofthe second size type from the 8-bit primary transform core matrix thatis 64-point or larger and cause or transmit information that causes thetarget file for encoding or decoding to be encoded or decoded using theextracted second transform core matrix of the second type, wherein whena result of the determination is to use the second transform core matrixof the second size type that is smaller than the first size type:extracting the second transform core matrix of the second size type fromthe 8-bit primary transform core matrix that is 64-point or larger andcausing or transmitting information that causes the target file forencoding or decoding to be encoded or decoded using the extracted secondtransform core matrix of the second type, and wherein when the firsttransform core matrix is a 64-point, 8-bit primary transform corematrix, which is constructed using thirty-three integers which are usedto construct a 32-point, 8-bit primary transform core matrix and anadditional thirty-two integers that include: bf, bg, bh, bi, bj, bk, bl,bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb, cc, cd,ce, cf, cg, ch, ci, cj and ck plus their respective sign changes, andthe additional thirty-two integers {bf, bg, bh, bi, bj, bk, bl, bm, bn,bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb, cc, cd, ce, cf,cg, ch, ci, cj, ck} are equal to {91, 90, 90, 90, 88, 87, 86, 84, 83,81, 79, 77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44, 41, 37, 33, 28, 24,20, 15, 11, 7, 2}.
 13. The apparatus according to claim 12, wherein thefirst primary transform core matrix is a Versatile Video Coding (VVC)standard transform.
 14. The apparatus according to claim 12, wherein thesecond primary transform core matrix is one of DCT-8, DST-1 or DCT-5.15. The apparatus according to claim 12, wherein when deriving theN-point 8-bit DCT-8 primary transform core matrix or the N-point 8-bitDST-7 transform core matrix, the set of M unique numbers which are usedto construct the N-point DCT-8 core matrix is the same set of M uniquenumbers which construct the N-point DST-7 core matrix.
 16. The apparatusaccording to claim 15, wherein M equals to N.
 17. The apparatusaccording to claim 12, wherein a left half or a right half of even orodd rows of the 8-bit primary transform core matrix form a matrix whichis similar to the second transform core matrix.
 18. The method accordingto claim 17, wherein the 8-bit primary transform core matrix is one of:DCT-2, DCT-5, DCT-8, DST-1 and DST-7, and the second transform corematrix is one of: DCT-2, DCT-5, DCT-8, DST-1 and DST-7.
 19. The methodaccording to claim 18, wherein the 8-bit primary transform core matrixand the second transform core matrix are the same one of: DCT-2, DCT-5,DCT-8, DST-1 and DST-7.
 20. A non-transitory computer-readable storagemedium storing instructions that cause one or more processors to:receive information regarding a target file for encoding or decoding,the target file for encoding or decoding being one of: a compressedvideo or image data file or an uncompressed video or image data file;determine, for the encoding or decoding of the target file, whether touse a first transform core matrix that is of a first size type or asecond transform core matrix that is of a second size type that issmaller than the first size type, wherein the first size type is the8-bit primary transform core matrix, which has the core matrix that isrepresented by 8-bits and that has a size that is 64-point or larger;when a result of the determination is to use the first transform corematrix of the first size type, cause or transmit information that causesthe target file for encoding or decoding to be encoded or decoded usingthe 8-bit primary transform core matrix that has the size that is64-point or larger; and when a result of the determination is to use thesecond transform core matrix of the second size type that is smallerthan the first size type: extract the second transform core matrix ofthe second size type from the 8-bit primary transform core matrix thatis 64-point or larger and cause or transmit information that causes thetarget file for encoding or decoding to be encoded or decoded using theextracted second transform core matrix of the second type, wherein whena result of the determination is to use the second transform core matrixof the second size type that is smaller than the first size type:extracting the second transform core matrix of the second size type fromthe 8-bit primary transform core matrix that is 64-point or larger andcausing or transmitting information that causes the target file forencoding or decoding to be encoded or decoded using the extracted secondtransform core matrix of the second type, and wherein when the firsttransform core matrix is a 64-point, 8-bit primary transform corematrix, which is constructed using thirty-three integers which are usedto construct a 32-point, 8-bit primary transform core matrix and anadditional thirty-two integers that include: bf, bg, bh, bi, bj, bk, bl,bm, bn, bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb, cc, cd,ce, cf, cg, ch, ci, cj and ck plus their respective sign changes, andthe additional thirty-two integers {bf, bg, bh, bi, bj, bk, bl, bm, bn,bo, bp, bq, br, bs, bt, bu, bv, bw, bx, by, bz, ca, cb, cc, cd, ce, cf,cg, ch, ci, cj, ck} are equal to {91, 90, 90, 90, 88, 87, 86, 84, 83,81, 79, 77, 73, 71, 69, 65, 62, 59, 56, 52, 48, 44, 41, 37, 33, 28, 24,20, 15, 11, 7, 2}.